tag:blogger.com,1999:blog-67088970415299045002024-03-05T15:39:22.375-06:00RATIONAL LIMITSSlowly becoming a better teacher.Unknownnoreply@blogger.comBlogger25125tag:blogger.com,1999:blog-6708897041529904500.post-45580023006865199432014-05-07T09:18:00.000-05:002014-05-07T09:18:08.860-05:00Moving blog!Hi all,<div>
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My blog has moved to <a href="http://kevinkrenz.com/">kevinkrenz.com</a>. Also, the RSS feed has changed since I moved my site - please update that if you are still interested!</div>
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Kevin</div>
Unknownnoreply@blogger.comtag:blogger.com,1999:blog-6708897041529904500.post-8360927081396876152013-05-05T18:57:00.001-05:002013-05-05T21:12:01.405-05:00Motivating trig through modeling<span style="font-family: inherit;">I'm starting to love teaching trig. In the past few terms, we've ended the unit with a <a href="http://blog.kevinkrenz.com/2012/08/nbi-modeling-temperatures-with-geogebra.html" target="_blank">GeoGebra project where we model monthly temperatures in Wisconsin</a>. The project is a great way to see if students have connected the ideas to the real world and the students actually end up thinking it's a little cool.</span><br />
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<span style="font-family: inherit;">However, I don't like having to do a week or two of abstract, context-free trig before really getting to students to see why it matters. So, to try something a bit different, I put together an intro task where they look at some more real data - in this case, a fellow teacher's monthly gas bill. You can find that task <a href="https://docs.google.com/document/d/1LVUwzCIuHzZYOkH4MnNtE6N5DXdxpub6h1WcpPBlYxg/edit" target="_blank">here</a>. I give them this graph and a few questions:</span><br />
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<span style="font-family: inherit; margin-left: 1em; margin-right: 1em;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBUQG7htCplK2ncIYdwSPIzI1KYbUuAa22Tjwh-dtEv6XQS_8q_HDOVGQG1I9PUqCshiaI8VwPyEzr53t6e17_xfsBndKIplQVlhLfg1RyEdrqGGzngWGm26uHF8yJqld76Qoac5-7ULza/s1600/Screen+Shot+2013-05-05+at+5.12.42+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="246" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBUQG7htCplK2ncIYdwSPIzI1KYbUuAa22Tjwh-dtEv6XQS_8q_HDOVGQG1I9PUqCshiaI8VwPyEzr53t6e17_xfsBndKIplQVlhLfg1RyEdrqGGzngWGm26uHF8yJqld76Qoac5-7ULza/s400/Screen+Shot+2013-05-05+at+5.12.42+PM.png" width="400" /></a></span></div>
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<b id="docs-internal-guid-6c4e9a2b-770e-112d-b9f2-267bd9c1c5ef" style="font-weight: normal;"><span style="font-family: inherit;"><i style="font-style: normal;"></i></span></b></div>
<ol style="margin-bottom: 0pt; margin-top: 0pt;">
<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<b id="docs-internal-guid-6c4e9a2b-770e-112d-b9f2-267bd9c1c5ef" style="font-weight: normal;"><span style="font-family: inherit;"><i><span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">How many years of data is represented on the graph? How do you know?</span></i></span></b></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<b id="docs-internal-guid-6c4e9a2b-770e-112d-b9f2-267bd9c1c5ef" style="font-weight: normal;"><span style="font-family: inherit;"><i><span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Circle the point associated with the most expensive bill.</span></i></span></b></div>
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<b id="docs-internal-guid-6c4e9a2b-770e-112d-b9f2-267bd9c1c5ef" style="font-weight: normal;"><span style="font-family: inherit;"><i><i><ol style="margin-bottom: 0pt; margin-top: 0pt;">
<li dir="ltr" style="font-size: 15px; list-style-type: lower-alpha; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">How much did he pay?</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: lower-alpha; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Which month of the year do you think that was? Why?</span></div>
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<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Put a box around a point associated with an “average” bill.</span></div>
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<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">How much did he pay?</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: lower-alpha; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Which month of the year do you think that was? Why?</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">On the graph, sketch a curve that fits the data.</span></div>
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<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Let’s convert that graph into a different form. Create a table for months 6 - 29.</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">How would you describe the pattern of the gas bills?</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Predict what Mr. Bruns’ gas bill will be in the 40</span><span style="font-size: 10px; vertical-align: super; white-space: pre-wrap;">th</span><span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;"> month. Show or explain your reasoning.</span></div>
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<li dir="ltr" style="font-size: 15px; list-style-type: decimal; vertical-align: baseline;"><div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;">Predict what Mr. Bruns’ gas bill will be in the 100</span><span style="font-size: 10px; vertical-align: super; white-space: pre-wrap;">th</span><span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;"> month. Show or explain your reasoning.</span></div>
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</i></span></b></ol>
<div dir="ltr" style="line-height: 1.15; margin-bottom: 0pt; margin-top: 0pt;">
<b id="docs-internal-guid-6c4e9a2b-770e-112d-b9f2-267bd9c1c5ef" style="font-weight: normal;"><span style="font-family: inherit;"><i><span style="font-size: 16px; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">Hmm, it would be awfully nice if we had a </span><span style="font-size: 16px; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">function</span><span style="font-size: 16px; font-weight: bold; vertical-align: baseline; white-space: pre-wrap;"> that could do some of the work for us...</span></i><span style="font-size: 15px; vertical-align: baseline; white-space: pre-wrap;"></span></span></b></div>
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<span style="white-space: pre-wrap;">From here, I'm thinking that we would go on to develop a "cost" function, connect it to a circle, and then get to a point where there is a motivation to develop the unit circle, sine, and cosine:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9XtEq4NdRswxVdD9AFdb_mmeF3daCZFgQlhveO9Ac-gPhRZC2EujpIqnyAJbMpbBtVz-1_LJgMCy2yydcJTeKaIECWK4LOJzOWvrtESTcGpkJiD0nXpohhqzGQN4hO3p9jlwOH4O8Ra9H/s1600/IMG_20130505_184344.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9XtEq4NdRswxVdD9AFdb_mmeF3daCZFgQlhveO9Ac-gPhRZC2EujpIqnyAJbMpbBtVz-1_LJgMCy2yydcJTeKaIECWK4LOJzOWvrtESTcGpkJiD0nXpohhqzGQN4hO3p9jlwOH4O8Ra9H/s320/IMG_20130505_184344.JPG" width="147" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisG4GcQbr96LsMzYR6OHqYipAe27xgPj09kCLxtRf7pBGywrn6bO9hpmkghq-89cRfV8EVxm0OukaZUpIP64Du9uMpAw-tzPRp3YQMNHjoYYnztI-g4dDVjkT8iYvIJ7gOB8grPOpa7fbD/s1600/IMG_20130505_184434.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisG4GcQbr96LsMzYR6OHqYipAe27xgPj09kCLxtRf7pBGywrn6bO9hpmkghq-89cRfV8EVxm0OukaZUpIP64Du9uMpAw-tzPRp3YQMNHjoYYnztI-g4dDVjkT8iYvIJ7gOB8grPOpa7fbD/s320/IMG_20130505_184434.JPG" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhA8z2KShB9NugTgqW3RvoaAI3l2nyaNGnSlDUIebx68xPWqkvMtzhRDCD2oaTmDMQljIBeSxHJSf5umHGfT6FDoHNCHOLNGxAUPcgzOKd4uIbFtTZITbgErIEcWtVgFelgbeQ1Cmo6oAi8/s1600/IMG_20130505_184409.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhA8z2KShB9NugTgqW3RvoaAI3l2nyaNGnSlDUIebx68xPWqkvMtzhRDCD2oaTmDMQljIBeSxHJSf5umHGfT6FDoHNCHOLNGxAUPcgzOKd4uIbFtTZITbgErIEcWtVgFelgbeQ1Cmo6oAi8/s320/IMG_20130505_184409.JPG" width="314" /></a></div>
<span style="white-space: pre-wrap;">I'm going to try this out this week - we'll see how it goes!</span></div>
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<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-75418866257837898422012-12-16T19:42:00.001-06:002012-12-16T19:42:46.013-06:00Algebra 2 & relationshipsWe've been working on our algebra 2 curriculum over the past year, both to align it with the Common Core and make it seems less like a random collection of function things. Last year we had the idea to make it very data-based; the idea was to have students gather a ton of data, and then base our study of functions on those data sets.<br />
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There are a few problems with this approach. For one, our district decided that prep periods are a luxury that cannot be afforded, so we don't have the time to prepare such a radical departure from our current curriculum.<br />
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Second, and more problematically, I think this would put way too much emphasis on the applied part of mathematics, and would make it difficult to meaningfully teach any aspects that aren't as practical. If we're motivating the study of functions based on answering questions about real data, I'm not sure how I'm going to help students understand removable discontinuities. Or motivate them to solve difficult equations algebraically when a graphical method yields a solution quickly. And so on.<br />
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Mathematics is not only about answering practical questions about the real world and data. It is certainly useful for that, but it's also much more. Thus a solely data-based approach to teaching algebra 2 would be a mistake, in my opinion.<br />
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So we need a more flexible approach if we're going to be true to all of mathematics. Thinking about the essence of our algebra 2 course, and really algebra in general, what we're dealing with is relationships. Linear, polynomial, rational, radical, exponential, logarithmic, and trigonometric, to be specific. We learn about these relationships by looking them in different lights - algebraically, graphically, and numerically.<br />
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Currently, a major problem with algebra 2 is that, to students, it feels like a ridiculously long laundry list of things to learn. I get the feeling that they don't understand that the equations, graphs, and tables that we work with <i>represent the same relationships</i>. I try to teach the connections, but they're mostly lost on students. And it's not surprising - the focus of the course isn't on understanding the relationships, it's on the set of techniques and facts about them. Without understanding the relationships themselves.<br />
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So, my idea: refocus the course on the relationships. One way to do that might be to start numerically. For example, we might start with the following:<br />
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<b>{ </b><b>(1, -3), </b><b>(10, 1019), </b><b>(-1, -4.5), </b><b>(?, 59),</b><b> (2, -1), (3, 3), </b><b>(0, -4),</b><b> (0.5, ?) }</b></div>
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I think this would really put the focus on the relationship itself. We're dealing with two quantities, and there is some way to predict one from the other. However, it is difficult to figure that out simply with numbers, so translating the problem into different forms - equations and graphs - takes on more meaning. </div>
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Our new semester starts in about a month; I'll let you know how it goes. Any wisdom you have to offer would be appreciated.</div>
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<br />Unknownnoreply@blogger.com5tag:blogger.com,1999:blog-6708897041529904500.post-13854901839301079952012-12-09T19:05:00.001-06:002012-12-09T19:25:42.493-06:00Rekindling why I teach mathematics<div class="separator" style="clear: both; text-align: left;">
I haven't blogged for a long time, and I'll explain that in relatively short time. But in the mean time, I have a brief thought to share.</div>
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I've really, really been struggling lately to find my passion for teaching pure mathematics. I'm very passionate about teaching computational thinking skills and statistics - ideas and methods that clearly are relevant to developing an informed citizenry - but I've been questioning the point of teaching apathetic teenagers rather abstract mathematics.</div>
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Well, I've found the beginning of a cure in, of all places, a graphic novel. This weekend I've been reading <a href="http://www.amazon.com/Logicomix-Search-Truth-Apostolos-Doxiadis/dp/1596914521">Logicomix</a>, a story about the life and ideas of Bertrand Russell.</div>
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<a href="http://upload.wikimedia.org/wikipedia/en/thumb/6/60/Logicomix_cover.jpg/250px-Logicomix_cover.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="http://upload.wikimedia.org/wikipedia/en/thumb/6/60/Logicomix_cover.jpg/250px-Logicomix_cover.jpg" width="228" /></a></div>
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For those of who don't know, or have forgotten, Russell's main role in the story of mathematics was his attempt to create a solid foundation of all of mathematics in his <a href="http://en.wikipedia.org/wiki/Principia_Mathematica">"Principia Mathematica"</a>, co-authored with Alfred Whitehead. This graphic novel does a beautiful job of connecting the ideas in mathematics to the questions that make our existence so wonderfully interesting.</div>
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I need to remember, I'm teaching ideas. And these ideas have a context; they are part of the story of real people attempting to more fully understand our place in the universe.</div>
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I haven't been teaching that at all.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-26921367871181430362012-09-04T19:41:00.005-05:002012-09-04T21:32:00.328-05:00[NBI] My teaching story<i><span style="font-size: x-small;">This post is part of the <a href="http://samjshah.com/2012/08/06/new-blogger-initiation-pledge-by-tuesday-august-14th/" target="_blank">new blogger initiative</a>.</span></i><br />
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Every teacher has a handful of characteristics that make him or her unique. While your teaching style may share characteristics with other teachers, it's the unique combination that makes you different. It's interesting to figure out where those come from.<br />
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Right before I started my first year of teaching, we did this activity:<br />
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<ol>
<li>Put a post-it note in the middle of a blank piece of paper.</li>
<li>Think of your four favorite teachers you've had.</li>
<li>Write their names on the paper around the edges of the post-it note.</li>
<li>Now think about each teacher individually. Choose one word that describes why you wrote that teacher's name.</li>
<li>Remove the post-it note.</li>
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Ta da! How similar does that look to your teaching style?</div>
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Here's mine:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWxAQx-OjfZMZuRT-it9L3po7G5wv9Mnn-J2CLlBJWNuhuUMkcnmK8Y6q2WbYzXElwRm1fwzZbIJx-suEeLvu7yO3BSyRhJjxyc6x6hbFEfZoxood11UV2DXO6BBxa1s1BEnOdt1srngIl/s1600/post-it.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWxAQx-OjfZMZuRT-it9L3po7G5wv9Mnn-J2CLlBJWNuhuUMkcnmK8Y6q2WbYzXElwRm1fwzZbIJx-suEeLvu7yO3BSyRhJjxyc6x6hbFEfZoxood11UV2DXO6BBxa1s1BEnOdt1srngIl/s320/post-it.jpg" width="240" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fun, faith, inquiry, & reach</td></tr>
</tbody></table>
<i>Fun (high school Spanish teacher)</i><br />
I wasn't convinced I wanted to be fluent in Spanish, but we had so much fun in this class that we didn't realize how much we were learning. I recognize that now with how much Spanish I still remember, even though its chief use is when we go out to eat.<br />
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<i>Faith (high school computer science teacher)</i><br />
He challenged us, a lot. But it was always obvious that he truly believed and knew that we could do what he threw at us. And you know what? We always did. After a while, I just started to figure that if he was assigning it, I could do it.<br />
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<i>Inquiry (high school physics teacher)</i><br />
This was my introduction to a truly inquiry-based modeling class. He drove a lot of my peers mad because he wouldn't just give us the answers, but he was one of the first teachers to introduce the idea that we're fully capable of creating our own knowledge.<br />
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<i>Reach (high school orchestra teacher)</i><br />
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I remember sitting in orchestra after a recent concert and getting the new music. We would look at it, try to play some of it, and sometimes barely make it through a measure. Follow this up with one of my most vivid memories from high school - the director tearing up on stage after finishing the last note. That's what I mean by reach - she showed us what we could do.<br />
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The really powerful part</h2>
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I had a student come in during lunch last year, and she saw the post-it note hanging above my desk. "What's that?", she asked. I briefly explained it to her.</div>
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Her response? "Yeah, that really does fit you."</div>
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Just think about that. The qualities of my favorite teachers not only influence how I like to teach, but <b>my students recognize them in my teaching</b>. These teachers were so amazing that they're impacting <i>my</i> students.</div>
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That's the kind of teacher I want to be.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-62960335412783695142012-08-30T20:45:00.001-05:002014-02-03T20:37:27.727-06:00HELP! (or, be careful what you wish for)You know how there's that saying, "be careful what you ask for - you might just get it?"<br />
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It just happened to me.<br />
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School starts on Tuesday, and my block algebra 2 class just got moved to the Mac lab. Every day. Instead of teaching in a normal, boring math classroom with 28 deskchairs, four off-white walls, and some corny math posters, I am going to be in a room with 30 gigantic iMacs, along with with a side room that has a half dozen circular tables with chair. I couldn't ask for a much better setup.<br />
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I have five days to figure out some kind of rough plan for how I'm going to approach and teach this class differently. Feel free to skip the next section if you don't care about what led to this.<br />
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The backstory</h2>
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I was frustrated last year with the limited access to technology that my classes I had, and it didn't look like it was going to get any better. So I decided to take matters into my own hands, and I started collecting old computers from friends and family. I cleaned them up, installed <a href="http://www.ubuntu.com/">Ubuntu</a> or <a href="http://lubuntu.net/">lubuntu</a>, along with <a href="http://geogebra.org/cms/">GeoGebra</a>, <a href="http://scratch.mit.edu/">Scratch</a>, and <a href="http://matplotlib.sourceforge.net/">matplotlib</a> for <a href="http://www.python.org/">Python</a>.</div>
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That project is coming along. It would be great to have one or two more, but I should enough functional computers in my classroom for groups of four to be able to share a computer with GeoGebra, Scratch, and Python.</div>
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Then our tech person came into my classroom today. She asked a polite question about how my mini-lab was coming, and then said, "I don't want you to have to do this. I mean it's great that you are, but you shouldn't have to. The Mac lab is open block D - what class do you have then? Algebra 2? Ok, you're teaching that class in the Mac lab."</div>
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Umm, what? Really? This is going to take a day or two to sink in.</div>
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<br /></div>
<div>
I want to underscore, I am ecstatic about this. I'm not complaining. It's just that school starts in five days, and I don't want to squander this. There is so much that I could do - I just don't know what that is yet, or how I'm going to do it.</div>
<div>
<br /></div>
<h2>
Ideas</h2>
<div>
I don't want to just list the specific tools that I'm going to use. I want to develop a picture of what I want my students doing that is different than the standard paper-and-pencil based algebra 2. Then I'll figure out what programs, services, whatever that will help us accomplish that.</div>
<div>
<br /></div>
<div>
So, my list:</div>
<div>
<ul>
<li>Work with large data sets</li>
<li>Programming</li>
<li>Researching real problems</li>
<li>Presenting with digital resources</li>
<li>Writing</li>
<li>Learning to become self-directed learners</li>
<li>Live backchanneling</li>
<li>Doing all of the above collaboratively</li>
</ul>
<div>
I could probably add more, but I think that's a decent set of ideas. I'll certainly try to give computational thinking a central role in the class.<br />
<br />
Where I need help is making this happen in algebra 2. How can I adapt our unit on polynomials to a classroom with this kind of access to technology? What can I do with solving radical equations?</div>
</div>
<div>
<br /></div>
<div>
If you have any ideas, please comment, <a href="https://twitter.com/kevin_krenz">tweet</a>, <a href="https://plus.google.com/100907417191360907817/posts">Google+</a>, or <a href="mailto:kevin.krenz@gmail.com">email</a> me. I promise to share what I do and give credit where it's due. The six units we teach are polynomials, rational functions, radical functions, exponential & logarithmic functions, trigonometric functions, and statistics.</div>
<div>
<br /></div>
<div>
Seriously. Whether it's just an idea you have or a lesson you've taught ten times, if you're reading this, please share. I've been given the blessing/curse of what I wanted, and I want to make sure I actually take advantage of it.</div>
<div>
<br /></div>
<div>
Please.</div>
Unknownnoreply@blogger.com7tag:blogger.com,1999:blog-6708897041529904500.post-43809045744647717992012-08-25T09:13:00.001-05:002012-08-25T09:13:36.936-05:00Transformations with Scratch<h2>
Students need to do more</h2>
I don't mean problems. I mean building. Creating. Writing. Designing. I think it can be amazing [read: tragic] how little students actually produce in a typical math class - no wonder math is often perceived as abstract and disconnected from the real world.<br />
<br />
Having students do more is good for about a million reasons. They provide a firm foundation to build on their prior knowledge and experience, formulate questions, climb the <a href="http://blog.mrmeyer.com/?s=%5BLOA%5D">ladder of abstraction</a>, and experience how math is connected to the world they live in. And the best part? The result of their growth is concrete. It's not a completed worksheet, or a good grade on a test. Maybe it's a video, or a presentation, or a programmed simulation, or whatever. The point is that it is something much more real.<br />
<br />
<h2>
Challenges</h2>
<div>
That all being said, there are reasons why projects aren't the norm in math. I know some schools have taken bold steps and are letting going of pre-determined curricula in place of project-based learning and student-driven inquiry, but they're the minority. And not my school.</div>
<div>
<br /></div>
<div>
In a typical public school, I think these are the main challenges for using projects in math: </div>
<br />
<ol>
<li>Finding projects that fit the unit.</li>
<li>Student expectations of what math class is.</li>
<li>Presenting the projects to students.</li>
<li>Assessing the projects.</li>
<li>Instructional time.</li>
<li>Planning time.</li>
</ol>
<div>
My plan is not to solve all of these challenges this year. My hope is to create some space and a framework that allows me to insert projects more frequently.</div>
<div>
<br /></div>
<h2>
First project: Transformations with Scratch</h2>
<div>
I participated in a <a href="http://www.cs4hs.com/index.html">CS4HS workshop</a> this summer and was introduced to <a href="http://scratch.mit.edu/">Scratch</a>. The best way to learn what it is is to play with it. Go ahead, it's a free download, cross-platform, and fun.</div>
<div>
<br /></div>
<div>
The first project of the year for my geometry students is going to be creating a 30 second animation with Scratch. I like this it introduces programming as well as gives students a deeper experience with transformations. </div>
<div>
<br /></div>
<div>
Here's the handout for students that I will use. The front side presents the task and details to the student; the back contains the rubric by which the projects will be assessed.</div>
<div>
<br /></div>
<a href="http://www.scribd.com/doc/103906726/Scratch-project" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Scratch project on Scribd">Scratch project</a><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772727272727273" data-auto-height="true" frameborder="0" height="600" id="doc_809" scrolling="no" src="http://www.scribd.com/embeds/103906726/content?start_page=1&view_mode=scroll&access_key=key-2nfbtd7kir6mpmeayyx3" width="100%"></iframe>
<br />
<div>
<br /></div>
<div>
I could greatly use some critical feedback before I give this a go. Design, language, rubric, whatever. The "product" part of the rubric definitely needs improvement, but I'm having trouble describing what it is that I want students to produce.<br />
<br />
After this project, I plan on tweaking the setup and then using this as my template for the rest of the projects. Hopefully it will making creating new ones quicker and easier!</div>
<div>
<br /></div>
<div>
<br /></div>
Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-6708897041529904500.post-14425772353394727782012-08-22T11:48:00.000-05:002012-08-22T17:43:55.347-05:00[NBI] Modeling Temperatures with GeoGebra<i><span style="font-size: x-small;">This post is part of the <a href="http://samjshah.com/2012/08/06/new-blogger-initiation-pledge-by-tuesday-august-14th/" target="_blank">new blogger initiative</a>.</span></i><br />
<br />
The more that I grow as a teacher, the more I believe in teaching modeling with mathematics.<br />
<br />
That might sound somewhat silly. I've always known, in some sense, that modeling is a good idea. They certainly encouraged it in my prep program and it's a significant part of the Common Core. However, when you're actually making instructional decisions for the day or week, modeling can be tough to choose. It takes more class time, and in some ways, can seem "softer".<br />
<br />
But every time I have my students complete a good modeling activity, those thoughts just seem ridiculous. I see real struggle. I see connections get made and broken. I see students conjecture. I see them contextualize and decontextualize. Beyond the truly deep learning, I haven't encountered a better assessment tool.<br />
<br />
<h2>
Monthly temperatures in Wisconsin</h2>
One of the better lessons that I taught during my first year of teaching was a modeling lesson. We were finishing up a unit on trig functions and I *thought* that my students had learned the transformations. I wanted them to see that what we learned really does show up in the world, so I grabbed the last five years of monthly temperatures in Wisconsin from the <a href="http://www.ncdc.noaa.gov/oa/climate/climatedata.html#monthly">Online Climate Data Directory</a>. I then entered that data in the spreadsheet of GeoGebra, created a list of points, and distributed the GeoGebra file to my student. (I would love for my students to be able to handle the data, but I made an executive decision based on time-constraints).<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNra4DpRfti_Llj1yTmXtnLIzxxmUnSkskdSPqERPogdeYOm256LdEIkRijUcJijOy3hijc82fQhyYBCRU5msZVSQfFzYYCl9hXqzWsdEfPlL0sgICjy-Jeupf1nGkDvdH12oXLdNSHvok/s1600/Screen+Shot+2012-08-22+at+11.13.14+AM.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNra4DpRfti_Llj1yTmXtnLIzxxmUnSkskdSPqERPogdeYOm256LdEIkRijUcJijOy3hijc82fQhyYBCRU5msZVSQfFzYYCl9hXqzWsdEfPlL0sgICjy-Jeupf1nGkDvdH12oXLdNSHvok/s400/Screen+Shot+2012-08-22+at+11.13.14+AM.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">I know that it's supposed to look like this, but it still kind of amazes me. </td></tr>
</tbody></table>
For the activity itself, I distributed the following halfsheets to guide students through the techy aspects of the activity:<br />
<a href="http://www.scribd.com/doc/103583302/Temperature-Directions" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Temperature Directions on Scribd">Temperature Directions</a><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772727272727273" data-auto-height="false" frameborder="0" height="800" id="doc_16572" scrolling="no" src="http://www.scribd.com/embeds/103583302/content?start_page=1&view_mode=scroll&access_key=key-2ge2lx1bv6cjtz8cj561" width="600"></iframe>
<br />
<br />
Basically, I wanted students to find an equation that fits the data as accurately as possible, interpret the different parts of their equation in the context of monthly temperatures, and present their work in a professional looking document. As we had done very little work of this type, I created a template with Google Docs to use.<br />
<a href="http://www.scribd.com/doc/103583304/Temperature-in-Wisconsin" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Temperature in Wisconsin on Scribd">Temperature in Wisconsin</a><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772727272727273" data-auto-height="false" frameborder="0" height="800" id="doc_92050" scrolling="no" src="http://www.scribd.com/embeds/103583304/content?start_page=1&view_mode=scroll&access_key=key-t14b8fl4o0bic9ch6mb" width="600"></iframe><br />
<br />
<h2>
Results</h2>
<div>
It wasn't perfect, but it was good. The insights into students' thinking were amazing. For example, it turned out to be very difficult for my students to interpret the period as one year, or that the midline represents the average temperature. I had a number of very good conversations trying to help students make these connections.</div>
<div>
<br /></div>
<div>
Additionally, I really enjoyed watching students while they were fitting equations to their data. I had considered setting up the equation for them with sliders, but I'm glad that I didn't. There was value in having them manually type in and change constants in the appropriate places in their equation. Students were checking their notebooks to figure out how to change the amplitude, or just tinkering to see if it worked. Really, either was fine with me. </div>
<div>
<br /></div>
<div>
In the end, I hope trig was a bit more concrete, and I had a much better idea of what my student understood.</div>
<div>
<br /></div>
<h2>
Next time</h2>
<div>
Instead of ending the unit with this, I will start with it. Trig would then come from the real world and students would be playing with all of the transformations before I teach the vocabulary. We could then discuss the ideas of trig transformations with a solid context to refer to and draw comparisons to other places in the world - "How do you think the amplitude of the monthly temperatures in Wisconsin compares to Cuba's? Why? What about the period?"</div>
<div>
<br /></div>
<div>
Next time.</div>
Unknownnoreply@blogger.com6tag:blogger.com,1999:blog-6708897041529904500.post-91800615523252409272012-08-20T09:59:00.000-05:002012-08-22T11:49:19.065-05:00Geometry summer curriculum workThis summer our department was given some curriculum hours to develop/reshape our geometry courses in light of the Common Core State Standards (CCSS). I'll assume that you know something about this - I just want to share the process we went through and where we are currently.<br />
<br />
At the beginning of the summer, we had a consultant give a short workshop for math teachers in our district. To be perfectly honest, I didn't find it incredibly helpful - possibly because I'm not that long removed from my certification program, where the CCSS were the only standards that we worked with.<br />
<br />
But, I did take one quite useful idea from the workshop - a process for breaking down the standards into something that we can get a better grasp on. I turned that process into a sensibly designed <a href="https://drive.google.com/previewtemplate?id=1LZ6Axccd17NUIzKXcdqbCAk27ibN9y4Kvkze9scQApM&mode=public">Google Docs template</a> to facilitate our summer work.<br />
<br />
So how did this work? Well, as a starting point, we decided to base all of our work of the suggested "pathway" from<a href="http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf"> appendix A of the CCSS for mathematics</a>. Within each unit, we looked at the different clusters. We decided that, for the most part, we're going to teach mini-units based on clusters.<br />
<br />
For example, here is the first cluster:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS7n6HA5P0JkWRTHbJinYQPADWtddFnFk3XMKRmUmlg-T_CMKZiP1K5lgLynwQSE57gmFN4H1ZEsBf5dp3XUlTzFMowdVqlGWIx2xN_mBNXS50l2xJ_cTBMEY1XQ_cragP94mHjvtNBd84/s1600/Common+Core+State+Standards+Initiative+-+Mathematics+-+High+School-+Geometry+-+Congruence(1).png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS7n6HA5P0JkWRTHbJinYQPADWtddFnFk3XMKRmUmlg-T_CMKZiP1K5lgLynwQSE57gmFN4H1ZEsBf5dp3XUlTzFMowdVqlGWIx2xN_mBNXS50l2xJ_cTBMEY1XQ_cragP94mHjvtNBd84/s400/Common+Core+State+Standards+Initiative+-+Mathematics+-+High+School-+Geometry+-+Congruence(1).png" width="400" /></a></div>
<br />
Reading through this it makes sense, but this is where is becomes more real. How we turn this into what we're teaching? We considered just making these our direct standards, but there was some messiness there. Concepts overlap, others would be difficult to assess, and so on.<br />
<br />
This is where breaking them down became quite helpful. For each cluster, we created a document that broke down each standard into the <i>what</i> and <i>how</i>. Looking at these all together, then, we would create our standards. We used some standards pretty much directly. Others we would combine, tweak, and interpret. Some standards we decided were things we would do/teach in class, like G-CO.4, but we wouldn't make them standards for assessment.<br />
<br />
For the above cluster, <a href="https://docs.google.com/document/d/1yfboa5WMZKQ5BmxCNFySUnbxyF-cLXT5G4Ch0TymTM4/edit">this is how we broke the standards down</a>, and <a href="https://docs.google.com/document/d/1PlQnQRwpDQyx1YllkHemNi-ebmDP2c5b7iuL8HvF38c/edit">these are the standards we developed</a>:<br />
<br />
<ol>
<li><span style="font-family: inherit;"><b>Geometric definitions</b><br /><span id="internal-source-marker_0.8937665629200637"><span style="vertical-align: baseline; white-space: pre-wrap;">Identify & precisely define angle, circle, perpendicular, parallel, & line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.</span></span></span></li>
<li><span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;"><b>Transforming figures
</b></span><b id="internal-source-marker_0.8937665629200637" style="font-weight: normal;"><span style="vertical-align: baseline; white-space: pre-wrap;">Reflect, rotate, and translate figures.</span></b></span></li>
<li><span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;"><b>Identifying transformations
</b></span><b id="internal-source-marker_0.8937665629200637" style="font-weight: normal;"><span style="vertical-align: baseline; white-space: pre-wrap;">Identify sequences of transformations that carry a given figure onto another.</span></b></span></li>
<li><span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;"><b>Transformations as functions
</b></span><b id="internal-source-marker_0.8937665629200637" style="font-weight: normal;"><span style="vertical-align: baseline; white-space: pre-wrap;">Describe a transformation as a function in the coordinate plane.</span></b></span></li>
</ol>
<br />
Now these are standards that I'm more confident that I can teach and assess.<br />
<br />
<h2>
What we're currently working on & where we're going</h2>
We're done with everything above, and we're finishing writing small formative assessments for every standard. We're then going to pull them together, tweak them, and create the summative assessments. Our school implements a 80% summative, 20% formative grading policy, so we're somewhat forced to break things down like this. I probably wouldn't otherwise.<br />
<br />
This was all rather mundane work, and I have a hard time calling it "important", especially considering what we might have done with our time otherwise. However, whatever gripes you may have with national standards, and I have several, we are legally obligated to teach them, are students are going to be judged on how well they know them, and we're likely going to be judged on how well our students do. So, in some manner of speaking, this was important work to complete.<br />
<br />
There's clearly something missing from this model. It treats math as the sum of the standards, when it should be so much more than that. At the beginning of the summer, we were hoping to include a "cluster question" within every cluster of standards that would be an interesting problem, application, project, or whatever that uses the main idea of the cluster. We didn't have time to develop them, but those could help.<br />
<br />
Incorporating the <a href="http://www.corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/">practice standards</a> will help. I'm going to try to use them daily to guide my teaching. But what I really want to do is use this work as the base for turning geometry into a project-based course. We'll have a solid grasp of what our students need to learn, along with some of the how. With this foundation laid, I think we could do something really special while respecting our commitment to teach the standards.<br />
<div>
<br /></div>
<div>
<br /></div>
Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-6708897041529904500.post-15989879362717615922012-08-18T17:28:00.000-05:002012-08-22T11:49:33.173-05:00[NBI] Focus on culture<i><span style="font-size: x-small;">This post is part of the <a href="http://samjshah.com/2012/08/06/new-blogger-initiation-pledge-by-tuesday-august-14th/" target="_blank">new blogger initiative</a>.</span></i><br />
<br />
My primary goal for the first week of school is the same as it was last year - begin to build a classroom culture that supports exploration, creativity, safety, and fun. I'm hoping to fail less this year.<br />
<br />
Last year I failed for several reasons. First, I had no control over the physical environment. I was a traveling teacher without my own classroom, and the classrooms I was in were, well, boring. My wife is also a teacher, and we've sometimes discussed (read: grieved over) the fact that elementary school classrooms just look more inspiring that high school classrooms. Why is it that the typical second grade classroom looks like Google's offices and high school classrooms look like oversized cubicles? This year, <a href="http://host.madison.com/wsj/news/opinion/editorial/oconomowoc-worth-watching/article_8c96eb5c-9f9e-11e1-87d6-0019bb2963f4.html" target="_blank">thanks to a "bold" move by my school district</a>, I will have a classroom. So that will give me more control.<br />
<br />
However, that was far from the main reason that I failed. My chief failure last year was that I approached the building of a classroom culture as <i>solely my responsibility. </i>I didn't involve my students in its development. I planned activities, made seating charts, and listed important values, and then tried to project those onto my classes. In retrospect, I kick myself for thinking that I could build culture top down with any real success.<br />
<br />
It's especially annoying since this is a lesson I thought I had learned from observing our wars in Iraq and Afghanistan for the last ten years. Knowledge transfer is hard.<br />
<br />
The bottom line is that I cannot impose a culture on my classes. Culture is developed by groups of individuals who share common values and practices. I can help guide the development of our culture and be intentional about which values are emphasized, but I cannot create it on my own.<br />
<br />
So, concretely, what am I actually going to do? I've got about two weeks to figure that out. Here are my current ideas:<br />
<br />
<ol>
<li>Find a solid, collaborative, & creative task for my classes on the first day. I would like to begin our class with my students <i>doing</i> something</li>
<li>Figure out a way to spark a discussion around the ideas of collaboration, creativity, risk-taking, and fun. Round it out with an explicit mission statement of our class.</li>
<li>Have my student help design the physical setting of the classroom - decoration, seating arrangement, workspace design, etc.</li>
</ol>
<br />
I'd be very interested in hearing your ideas or words of wisdom!Unknownnoreply@blogger.com4tag:blogger.com,1999:blog-6708897041529904500.post-66961086741582538912012-08-11T21:26:00.000-05:002014-02-03T20:37:50.529-06:00Papert - What comes first, using it or 'getting' it?<blockquote class="tr_bq">
<i>"</i><span style="background-color: white;"><i>The principle is called the power principle or "what comes first, using it or 'getting it'?" The natural mode of acquiring most knowledge is through use leading to progressively deepening understanding. Only in school, and especially in SME is this order systematically inverted. The power principle re-inverts the inversion."</i></span> </blockquote>
<blockquote class="tr_bq">
<span style="background-color: white;">-Seymour Papert, <a href="http://www.papert.org/articles/AnExplorationintheSpaceofMathematicsEducations.html" target="_blank">An Exploration in the Space of Mathematics Education</a></span></blockquote>
A phrase I often hear is "teaching for understanding", which is a personal goal of mine. I don't just want my students to be able to <i>do</i>, I would like them to <i>understand</i>. So I try to introduce new material with exploratory activities where they wrestle with the material, apply what they already know, and try to create something new. My success varies, and I'm sometimes at a loss for why.<br />
<br />
It might be something as simple as this: I'm asking to students to understand something before they know what it is. I would imagine this is difficult indeed.Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-6708897041529904500.post-66802667925118175452012-08-02T10:03:00.000-05:002012-08-22T11:49:55.215-05:00Transforming geometryI never wanted to teach geometry. It bored me in high school, and I could never get myself particularly excited about the side splitter theorems or, well, anything about it. Ick.<br />
<br />
So naturally, geometry was and is one of my main course assignments as part of my first teaching position.<br />
<br />
Goal: make geometry interesting to teach and learn. Or, at least suck less.<br />
<br />
There are two reasons that this might actually happen - The <a href="http://www.corestandards.org/the-standards/mathematics">Common Core State Standards</a> and computational thinking.<br />
<br />
<h2>
Common Core</h2>
The first one is that the Common Core State Standards (CCSS) changed the focus of high school quite significantly by giving transformations a central role. Our department made the decision to embrace this direction and go with it. This summer we've been taking the suggested sequencing from <a href="http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf">Appendix A</a>, and then interpreting and translating this into a curriculum that we feel we can teach. If you want to a gander/copy/steal, our work is here (and still changing):<br />
<ul>
<li><a href="https://docs.google.com/document/d/1PlQnQRwpDQyx1YllkHemNi-ebmDP2c5b7iuL8HvF38c/edit">Unit 1: Congruence, proof, & constructions</a></li>
<li><a href="https://docs.google.com/document/d/1LHtORkZCpD6SGK1HiiGZPQKYuv1RiDs7GHq8nSFrdCM/edit">Unit 2: Similarity, proof, & trigonometry</a></li>
<li><a href="https://docs.google.com/document/d/1IXhoFYOvsGkTC5G4mgUgtdiCDn3bw-BbGjDb89rBRcI/edit">Unit 3: Extending to three dimensions</a></li>
<li><a href="https://docs.google.com/document/d/1Xw83Av5LDu7lJ9JdrC7xU8aiLEFiqP7PRwLwNfJ4k_4/edit">Unit 4: Coordinate geometry</a></li>
<li><a href="https://docs.google.com/document/d/1oKIivQBROfAqTiUx0J1vLLQyGhO1HUf_1HVClrJPOEY/edit">Unit 5: Circles</a></li>
<li><a href="https://docs.google.com/document/d/1m68wRbqmPbV0w_Rv9YqNh820ahuaGxQQQBXSbECkPNE/edit">Unit 6: Applications of probability</a></li>
</ul>
<div>
Now these alone aren't going to make geometry awesome, but I think that they do provide an improved sequencing and emphasis than a traditional geometry course. I'm also just getting a start on writing some assessments which is highlighting how different the approach is. I'll post more about that later.<br />
<br /></div>
<h2>
Computational thinking</h2>
<div>
This one has me much more excited. I already <a href="http://kevinkrenz.blogspot.com/2012/07/computational-thinking.html" target="_blank">wrote a post</a> about what it is and why I've decided to use it. At a high level, think of computational thinking as combining critical thinking with the power of computing, an incredibly relevant approach.<br />
<br />
Now that may not sound like it has much to do with high school geometry. High school geometry is traditionally a very abstract course where students are first introduced to mathematical rigor and proof. Unfortunately, we can fall into teaching geometry as a means to end (teaching proof), and we lose some of the utility and concreteness of geometry. After all, geometry is shapes!<br />
<br />
So with that motivation, I want to consistently incorporate computational thinking practices into geometry. I want to make geometry a more hands-on and practical course. I'll still teach proof & rigor, but I want students to associate geometry with geometry rather than proof.<br />
<br />
<br /></div>
Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-6708897041529904500.post-91571359907537621472012-07-26T15:40:00.001-05:002012-07-26T15:40:29.458-05:00Computational thinking<br />
<div style="margin-bottom: 0in;">
Now that I have a year under my belt, I
would like to think that I'm a little wiser. I know how to be really,
really inefficient while planning lessons. Goal number one of next
year: be a little less inefficient.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
A chief cause of the hours I would
spend planning a poor lesson was an overload of ideas and resources.
I would spend far too long reading through ideas, half starting, and
thinking. Then I'd run out of time, underplan an inquiry-based
activity, and run a bunch of copies of worksheets. During the lesson,
I'd get frustrated by the students' lack of initiative and interest
(caused by a lack of support and clarity).</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
And repeat, day after day. Lots of time
spent planning lessons that could have taken ten minutes to plan.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
I would like to be a little less
inefficient.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
My plan is to focus on one specific
framework. Instead of trying to do many different things because their cool, I want use a more focused set of ideas to both give my teaching more consistency and my planning more efficiency.</div>
<div style="margin-bottom: 0in;">
<h2>
<span style="background-color: white;"><br /></span></h2>
<h2>
<span style="background-color: white;">Framework: Computational Thinking</span></h2>
</div>
<div style="margin-bottom: 0in;">
<span style="background-color: white;">What is
computational thinking (CT)? Combine critical thinking with the power
of modern computing. This is not trivial – it is a significant
shift in how we develop students' ability to solve interesting
problems.</span></div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
There are several
ways of breaking down CT. <a href="http://www.google.com/edu/computational-thinking/">Google</a> and <a href="http://www.iste.org/learn/computational-thinking.aspx">ISTE</a> both offer definitions and resources. Some computer science curricula, <a href="http://www.exploringcs.org/">Exploring Computer Science</a> and <a href="http://www.csprinciples.org/">CS Principles</a> also offer approaches and resources, but are more computer science centric. The point is that there are
resources available and many smart people who think this is a
valuable approach.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
But, to make this
really clear, I think it's best to present an activity, and contrast a more traditional approach with a CT approach.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
<h4>
Activity:
Review solving linear equations</h4>
</div>
<div style="margin-bottom: 0in;">
<u>Traditional
approach</u>:
Model solving a few. Perhaps have students justify the steps to
underline the idea of equality. Then have them practice – either
alone or in groups. You could mix it up and play the mistake game if
you're feeling
different, or play a row game.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
<u>CT
approach:</u>
Write a linear equation on the board with one unknown. Have students
tell you how to solve it, step-by-step, but be an absolute pain in
the ass. Do only exactly what they say. Make them justify their
steps. Ask if you're done after every step. Do a couple of problems
like this.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Why? You're going to focus on solving linear equations as an
algorithm. Ask the students to write an algorithm to solve any given
linear equation. Explain what you mean by an algorithm – a very
precise step-by-step set of instructions that a trained monkey could
use to solve an equation.<br />
<br />
<a name='more'></a><br />
With this example, I'm guilty of something I find incredibly annoying - using the easy examples from algebra to demonstrate a different approach to teaching math. Sorry. However, I do plan on focusing largely on geometry this coming fall, which I anticipate to be one of the more difficult courses to switch to a computational thinking approach.</div>
<div style="margin-bottom: 0in;">
</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-56936934301443329382012-06-14T09:04:00.000-05:002012-06-14T09:04:43.652-05:00Teaching in Wisconsin<span style="font-size: large;">The Background</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlMfxI_e99arEIquC5lEzyAgQlrhnbJNqHlLp0qZBGW8UaIvS7j31wC03axDzrNCvmDQ3FYpNer_c2dTRu620OYoj4C01SLboKiiKFl6bGUqxmLTcfErSihGa6Om6w2bQ2UpSycKD50lZW/s1600/Split+Wisconsin.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlMfxI_e99arEIquC5lEzyAgQlrhnbJNqHlLp0qZBGW8UaIvS7j31wC03axDzrNCvmDQ3FYpNer_c2dTRu620OYoj4C01SLboKiiKFl6bGUqxmLTcfErSihGa6Om6w2bQ2UpSycKD50lZW/s320/Split+Wisconsin.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
In the past year and a half, teachers in Wisconsin have been in and out of the national spotlight after our governor's attack on collective bargaining rights. The bill sparked massive protests for weeks around the state capitol, and ultimately passed through a budgetary process. The law that was passed, known as Act 10, severely diminished the powers of unions and increased contributions to benefits. This was a leading cause of the unsuccessful recall election that took place recently.</div>
<br />
After Act 10 became law, there was a great deal of uncertainty about how districts would use their new powers. As it turns out, my district is going to be the first one to really put them to use and experiment. It was announced after spring break that in the future, nearly all high school teachers now are teaching the entire school with no prep period. In the core departments, this lead to approximately one out of every four teachers in the department being laid off. The remaining teachers will receive a bonus stipend of $14,000. In many respects, we now have a spotlight on us from across the state.<br />
<br />
<span style="font-size: large;">My Experience</span><br />
I don't write all of this to complain. I simply want to relate my experience of becoming a new teacher and completing my first year during this time as I think we are at a crossroads in education in Wisconsin, in more ways than one - how we teach, how we're judged, and what society believes about teachers.<br />
<br />
The climate has been decidedly negative. Very few people encouraged me to become a teacher, but more people than I count have questioned me<i>. </i>Or simply told me it's a poor choice. These voices have been inside and outside of the school, and the story they have been telling is the same: "You don't want to get on this train. It's breaking down and heading toward a cliff. There are better things to do with your life."<br />
<br />
Perhaps. There are changes coming, definitely. I doubt the wisdom of many of them, but I also know that this where I am supposed to be.<br />
<br />
Teaching is a strange profession. It seems that people either (a) think we're doing a truly noble service, or (b) think we're parasites who deserve less than what we earn. I have a lot of support - parents, students, friends, and community members who genuinely thank me for my daily work. And then I balance that with the looks of condescension and judgment, or people shouting "f@#$ you" out of their car window at me on my walk home because of a pin on my bag.<br />
<br />
At the end of the day, and the end of the year, I haven't decided how I feel or what I think. There's not a single way that I feel about teaching in Wisconsin right now. I love teaching - I know that - but teaching in Wisconsin right now is too complex to relate without several hours and a few pints. I don't know how long I'll be able to deal with the negativity and pressure.<br />
<br />
It's hard serving in a community that proudly voted nearly 3:1 for a governor that has extremely little respect for educators. I try to remind myself that I'm here for the students, but it is going to be my blood, sweat, and tears that make these new changes successful. What I fear is that my success is going to be used as proof of the validity of their ideas, even though they have handicapped me in an insulting manner.<br />
<br />
But that's not going to stop me, because I love this state. We have an amazing tradition of progressive thinking and high quality education (see: <a href="http://en.wikipedia.org/wiki/Wisconsin_Idea">The Wisconsin Idea</a>). Further, the people of this state are genuine, friendly, warm, and hard working, even if that doesn't seem to always be reflected in our politics. Politically, we are severely divided, and I am clearly in the slight minority. There are many, many people with us, but just a few more who disagree with us.<br />
<br />
I don't know what the future holds. I'm anxious about the teacher evaluation system that is now being piloted in a number of schools. I'm anxious that Wisconsin's commitment to a nation-leading public education system is slipping. I'm anxious about the quality of minds and characters who are going to volunteer to teach in this climate.<br />
<br />
I'll be here next year. I am committed to this state and want to believe that <a href="http://host.madison.com/ct/news/opinion/column/john_nichols/john-nichols-wisconsin-will-keep-bending-toward-justice/article_1b775ef8-af29-11e1-ac2a-0019bb2963f4.html">Wisconsin will keep bending toward justice</a> (great article by John Nichols). But I do have some serious doubts now, and that is new.Unknownnoreply@blogger.com8tag:blogger.com,1999:blog-6708897041529904500.post-55252835807265488572012-02-07T20:29:00.000-06:002012-02-07T20:34:35.696-06:00An equation is a problem...I have been teaching algebra 2 for about three weeks now, and I've noticed some serious deficiencies in my students' understanding of algebra. We have been reviewing the basics - solving linear equations and inequalities, graphing, and working with absolute value. Students are consistently making the same mistakes - their typical response is a vacant "oh", followed by a quick fix, and I'm struggling to help them understand what is happening.<br />
<br />
I've been scratching my head, rather anxious about our prospects of success. Our algebra 2 course is essentially a class about the major functions of mathematics - polynomial, rational, radical, logarithmic, exponential, and trigonometric. For each unit, we do the same set of things - graph, solve, and model.<br />
<br />
After working with them for a couple of weeks, I developed a suspicion that they don't really understand what we're working with - specifically, the equals sign. They can somewhat reliably graph and solve basic equations, but I've seen little evidence that they understand what these actually represent.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Equal_symbol.svg/500px-Equal_symbol.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Equal_symbol.svg/500px-Equal_symbol.svg.png" width="166" /></a></div>
<br />
<br />
So, I took five minutes from class yesterday and had them answer a few questions. Here are the questions with some typical responses:<br />
<br />
<ul>
<li>What is an equation?</li>
<ul>
<li>A problem that can be solved mathematically.</li>
<li>Something that has an end solution through multiple applications.</li>
<li>A problem you solve to get an answer.</li>
<li>It is a sequence of #'s and variables, and your job is to find the right answer to it.</li>
<li>Numbers or letters used to represent or solve something.</li>
</ul>
<li>What is a graph?</li>
<ul>
<li>A grid where you plot points.</li>
<li>A typically square thing you draw a line on.</li>
<li>Where points are plotted and to show increase and decrease.</li>
<li>A graph is a visual equation.</li>
<li>A graph is like another way of showing your work when you solve an equation.</li>
</ul>
</ul>
<br />
While there are some glimmers of hope in there, I think this is going to be a barrier for us. Students will not be able to reach the depth of understanding that I desire unless they understand what that damn equals sign actually means. And I think this boils down to looking at operators, relations, variables, constants, and expressions. Maybe not using that language, but building on those ideas.<br />
<br />
So, here's my plan. I'm going to students do an open sort with the following:<br />
<br />
<ul>
<li>Relations: =, <, >, <span style="line-height: 40px; text-align: -webkit-center;"><span style="font-family: inherit;">≤, </span></span><span style="line-height: 40px; text-align: -webkit-center;"><span style="font-family: inherit;">≥</span></span></li>
<li><span style="line-height: 40px; text-align: -webkit-center;"><span style="font-family: inherit;">Operators: +, -, </span></span><span style="background-color: white;">÷, </span><span style="background-color: white; line-height: 19px;"><span style="font-family: inherit;">×</span></span></li>
<li><span style="background-color: white; line-height: 19px;"><span style="font-family: inherit;">Variables: x, y, z</span></span></li>
<li><span style="background-color: white; line-height: 19px;"><span style="font-family: inherit;">Constants: 0, 1, 2, 3</span></span></li>
<li><span style="background-color: white; line-height: 19px;"><span style="font-family: inherit;">Expressions: 3x + 4, 5 - 1, 4ab</span></span></li>
<li><span style="background-color: white; line-height: 19px;"><span style="font-family: inherit;">Equations/inequalities: 2x </span></span><span style="background-color: white;">÷ 5 = 4, 2 - y < x, 5a <span style="line-height: 40px; text-align: -webkit-center;">≥</span> 8b, 1 + 1 = 2</span></li>
</ul>
<div>
After the items have been sorted, I'll encourage student to create a hierarchy of their groups. I really want to focus on the groupings - what do things have in common, how are they different, and what do they represent.</div>
<div>
<br /></div>
<div>
In the end, I'm hoping that my students have a better understanding of equality as a relationship, and that an equation is simply a statement that two expressions have the same value.</div>
<div>
<br /></div>
<br />
I would really appreciate any feedback before I give this a go!Unknownnoreply@blogger.com5tag:blogger.com,1999:blog-6708897041529904500.post-14531595878813327732012-01-17T21:05:00.001-06:002012-01-17T21:05:38.024-06:00The MulliganI don't love the block schedule. I'm not sure I even like it, and I'm not sure it's good for teaching math.<br />
<br />
But, as a first year teacher, there are two significant benefits:<br />
<br />
<ol>
<li>Reduced number of preps</li>
<li>I get a mulligan halfway through</li>
</ol>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibPLEYc8oxufn6EA0Z2ZnnnJ3kknJJifQCFFfGrPmZZkBmT36aZ3MsTpGvUpy6fzgEVwMU8auVRM3x2lnXcKpv2Wlk9kjkG649uH-K0vMrWeKgtU2jgpmCWiUS4xDqjpxockxEcFB3uI74/s1600/mulligan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibPLEYc8oxufn6EA0Z2ZnnnJ3kknJJifQCFFfGrPmZZkBmT36aZ3MsTpGvUpy6fzgEVwMU8auVRM3x2lnXcKpv2Wlk9kjkG649uH-K0vMrWeKgtU2jgpmCWiUS4xDqjpxockxEcFB3uI74/s320/mulligan.jpg" width="320" /></a></div>
<div>
<br /></div>
<div>
I'm pretty excited about #2 right now. Now that I have half a school year under my belt, there are about a million things I want to change, and some of the changes just work better when you get a new class. Here's a short list of changes I'm thinking about:</div>
<div>
<ul>
<li>Change the purpose of the openers</li>
<li>Start student portfolios</li>
<li>More mid-block formative assessments</li>
<li>Make whiteboarding and sharing an everyday part of class</li>
<li>Better cooperative group activities with individual accountability</li>
<li>More emphasis on communicating reasoning</li>
<li>Preparing for student mistakes instead of helping them avoid them</li>
<li>More consistency is the handling of assignments</li>
<li>Focus more on connecting with student</li>
<li>Focus less on the real word justification of the mathematics</li>
<li>Connect mathematical reasoning to students' lives more often</li>
<li>Follow through with consequences more consistently</li>
<li>Make better use of our weekly trips to the computer lab</li>
</ul>
<div>
The new term starts next Tuesday. While I'll certainly try to work on all of these, I'm going to pick two or three over the weekend to really set my sights on and reflect on in more detail. More to come later.</div>
</div>Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-6708897041529904500.post-56949990127760383442012-01-15T20:22:00.001-06:002012-01-15T20:22:47.362-06:00Becoming A Better MathematicianA year ago, there were two Project Euler problems that really gave me headaches (<a href="http://projecteuler.net/problem=26">#26</a> and <a href="http://projecteuler.net/problem=34">#34</a>, if you're really interested). I remember spending hours on them to no avail. This was frustrating, especially with how I work. I ended up skipping them and moving on to other problems, but that kind of thing doesn't sit well with me.<br />
<br />
So I took them up yesterday, and surprisingly, solved both without a terrible amount of trouble or time. I could chalk this up to simply taking a break and seeing the problem in a different light - which has worked for before - but I think it's more than that. Teaching high school mathematics has made me a much, much better mathematician.<br />
<br />
While this certainly gives me a personal sense of accomplishment, I'm more interested in the specific things that have made me a better mathematician and how I could use these in my teaching. Reflecting on my work yesterday, there were a few significant changes in the way I was working:<br />
<br />
<ul>
<li>Make big problems small</li>
<ul>
<li>My go-to approach is to plug some stuff in a small example, see how it works, and look for some kind of pattern.</li>
</ul>
<li>Slowing down</li>
<ul>
<li>I was solving for slowly - once I had an idea that I thought might work, I took the time to think/write out my reasoning <i>before</i> getting started on the solution.</li>
</ul>
<li>Starting over</li>
<ul>
<li>I was much more ok with completely scrapping something and starting in a new direction.</li>
</ul>
</ul>
<div>
<br /></div>
<div>
These are nothing revolutionary. Far from it. They are things I've heard and read before, and things I (sometimes) try to incorporate into my teaching. But there is something about experiencing this first-hand that really drives it home.</div>
<div>
<br /></div>
<div>
What's my take away? This <u>needs</u> to be a more conscious part of my teaching. It's not going to change a student in a day. It might now show up on a test. But at the end of the year, if I've done my job right, there will be a whole new level of problems that a student will now have the ability to take on. The process is slow, but without fruit.</div>
<div>
<br /></div>
<div>
I also need to keep doing mathematics. Whether it's inside the classroom or out, it's not an indulgence - it's a professional development necessity.</div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-29215619181761849622012-01-03T20:51:00.002-06:002012-01-03T20:51:38.011-06:00New Year's Resolutions<br />
<ol>
<li><b>Run</b> - three to four times a week. It makes me a happier, healthier person. I just need to commit to the time.</li>
<li><b>Read</b> - at least one book a month.</li>
<li><b>Blog</b> - one new post a week, even if I don't feel like I have much to say. Because when I start writing, I find that I always do.</li>
</ol>
<div>
<br /></div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-27070228484692736872011-12-28T21:00:00.000-06:002011-12-29T00:05:12.986-06:00School of One<div>
<div>
I've recently been asked to help develop a pilot "School of One" program in our district. For those of you not familiar with the project, take a look <a href="http://schoolofone.org/">here</a>. Basically, here's how it works.
<br />
<ol>
<li>Student comes into class and looks at their "playlist" - their options for what to do today. </li>
<li> Student makes their choice. This might be small group work, tutoring on a computer, direct instruction, etc. </li>
<li> Student gets started.</li>
<li> Student takes short formative assessment. </li>
<li> Teachers feed results into algorithm for all students. </li>
<li> Algorithm output lessons and activities to be prepared.</li>
</ol>
And repeat.
<br />
<br />
The more I think about School of One, the more ambivalent I am. There are some really amazing ideas and opportunities here, not the least of which is that students are given a higher degree of control and power over how they spend their time. To me, this is <b>the</b> benefit. There are others, to be sure, but I think that they all really revolve around this. Choice allows education to become more personal, more meaningful. Instead of being instructed exactly the same way as your other 29 classmates, your path is unique, based on who you are, what you're good at, and what you want to do.
<br />
<br />
I am excited to be involved, but not without some concerns and doubts. Rather than approach these as reasons not to help, I'm looking to these as the points I'll want to make sure are addressed by our work. It is the reason I am writing this post and the reason I hope you will help!
<br />
<br />
<em>Feedback and assessment</em>
<br />
As feedback has been one of the items I've been focusing on this year, this one concerns me. The only assessments built in are the short formatives given to student at the end of each lesson. The results are then fed back into the algorithm. If the student has demonstrated mastery, they move on. If not, they'll have to do a lesson based on the same objective again. Students involvement in this process is minimal, and there is essentially no feedback given to students. I don't like the idea that we just have a student spend another full lesson on the same content without any feedback other than the fact that they didn't get it yet. We need to do better than that.
<br />
<br />
<em>The algorithm</em>
<br />
This algorithm is basically supposed to work like Pandora or Amazon's suggested items. Based on your history, the available resources, and people like you, the algorithm suggests the lessons that will be the most productive for you. Without a doubt, a clever idea. There's a lot of faith placed in the algorithm, and yet remarkably little is shared and public about it. It was developed by a for-profity ed company, Wireless Generation, which is now owned by the News Corporation. That in itself scares me a bit, but I can get past that because we're almost certainly not going to be able to use it.<br />
<br />
What scares me about the algorithm is its potential to render a teacher's observations and professional judgment inconsequential. I do not want the role of the teacher to become something like a machine operator. The entire process needs to be within our control, which includes the inner workings of the algorithm. My fear is that the central piece of School of One, the algorithm, is a proprietary blackbox.<br />
<br />
The algorithm needs to be open source. It needs to be available to anyone and everyone, free, and without any restrictions. This process and how lessons are sequenced and chosen cannot be a proprietary secret if we really think that this model has the potential to change how we teach. Do not make this simply a tool that teachers use. Make it one they control.<br />
<em><br /></em><br />
<em>The big picture</em><br />
Each day's lessons are chosen based on a fairly specific roadmap of objectives that need to be covered. Even though the students path will be personalized, I'm a little anxious that we're not going to be taking students off of the path enough to really show them the world. Mathematics is much, much more than a series of facts and processes. It's a way of making sense of the world. I'm not sure how we're going to roll this in, but it will be a failure if we don't.
</div>
</div>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-67078729044347145442011-09-30T11:27:00.000-05:002011-09-30T11:27:11.978-05:00Sitting in the BackIn the last week or so, I've tried a new technique when I pose a question to the class, or want the class to solve a problem together: I sit in the back.<div>
<br /></div>
<div>
Typically, when I pose a question or a problem to the class, there are a few things that impede learning. First, the focus is still on me, so most of the students are just waiting for me to give the answer. Second, whenever a student does contribute an idea or question, it is directed at me. Not what I'm going for.</div>
<div>
<br /></div>
<div>
So, this week I wanted the class to work though a proof together. I drew the diagram on the board, asked the question, and then sat in the back row. This helped in two significant ways. For one, the focus was no longer on me; it was on the problem on the board. There was also a clear implication that with me <i>sitting</i> in the back - that this was <i>their</i> problem, and I'm giving them the time to do it (instead of the usual wait-long-enough-and-Mr.-Krenz-will-do-it-for-us game).</div>
<div>
<br /></div>
<div>
Additionally, the discussion of how to do the proof changed significantly. Questions and ideas were now being addressed to peers and the rest of the class rather than me. It wasn't me piecing together different students' correct responses to put a proof on the board with the false impression that the students did it. The students were offering ideas, critiquing, and collaboratively deciding on the next step.</div>
<div>
<br /></div>
<div>
When they completed a solidly challenging proof with almost no assistance, which they have been struggling with, I was still sitting in the back row. And I could not have been more proud of them.</div>
<div>
<br /></div>
<div>
It isn't perfect. I still give some guiding hints from the back, which I'm going to try to cut back on. The discussion can be confusing and solutions offered may not be correct. The path that is taken is not necessarily the most direct. But these are good things. Students start to understand that the perfectly linear proofs and thinking in the book isn't the way you need to think. I'm not sure it's the way that anyone thinks.</div>
<div>
<br /></div>
<div>
I'm sure other teachers have tried this. If so, I'd love to hear your experiences. If not, try it. Sit in the back and let me know what you think.</div>
Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-6708897041529904500.post-85490313410780006912011-09-22T14:37:00.000-05:002011-09-22T16:37:11.421-05:00What Do We Change?<i><span class="Apple-style-span" style="font-size: x-small;">This is long, and I'm unapologetically not going to bother proofreading much. This post is more or less a free write reflecting on the most intense day of my teaching yet. A day in which I didn't teach any math. I wrote this for myself, but I'd love to get some feedback.</span></i><br />
<br />
This is my first year teaching, and my first experience teaching geometry. I have three standard, blocked geometry classes, and it is hard. The pace is quick, most of my students are freshmen, and we're having trouble "covering" what we need to at a pace necessary to finish.<br />
<br />
I introduced proofs for the first time about a week ago, and we had a test yesterday. Scores were, well, less than what I was hoping for. I expected trouble with the proofs, which I found, but I was more concerned with the basic lack of understanding that I was seeing. The core pieces of the test - conditionals, converses, inverses, contrapositives, biconditionals, and the properties of congruence and equality - were simply not there. It's not that the students could properly interpret them and use them to judge and create. They failed to demonstrate a basic recall and application.<br />
<br />
As a new teacher, this is difficult to deal with. It's frustrating and depressing. Without a doubt, part of this reflects my (in)ability to effectively teach the material and skills. I think this is more devastating to me than it is to my students.<br />
<br />
However, I also know that this isn't all me. For one, I know that my students need to take responsibility for part of this. The choices that they're making in class aren't often the ones that will help them learn, and I think I need to teach that. Learning does not just happen; it is a choice that you make. I will do everything I can to make that a positive and helpful experience if you do choose to learn, but I can only do so much. I need to help my students understand that they are more of the class than me.<br />
<br />
So, the response. Even though our schedule is a breakneck pace, I decided to take a full block - 90 minutes - to talk about <i>how we do class</i>. Preparation, learning, studying, review, interaction, behavior, everything. I wanted to do something to help them realize <i>why</i> we are in class, what the goals are, and what drives my decision making process.<br />
<br />
So, rough lesson plan:<br />
<br />
<i>Mini-quiz</i><br />
I asked six very basic questions that only require remembering the information absolutely necessary to doing well on the test. This is basically the lowest bar. I then read off the answers, had students mark their own answers, and then asked them to think about how they did. We talked about Bloom's taxonomoy and the different levels of thinking.<br />
<br />
<i>Passed back tests</i><br />
I gave student a few minutes to go through their tests, thinking about what we had done so far. I explained that most scores were 5's and 6's. I use an 8-point standards-based grading system. A 5 means you demonstrated only a slight understanding of the concept, and a 6 means you're on the right track, but there is still a significant gap in your knowledge. So we have a ways to go yet.<br />
<br />
<i>Free write</i><br />
I gave students about 4-5 minutes to write whatever they were thinking and feeling. Frustration, anger, reflection, ideas for improvement, whatever. I said that I wouldn't collect, so they could write anything. During the third block, I gave the option to turn it in, with or without a name, and I was surprised at how many students <i>wanted</i> to turn it in. They felt a need to be heard.<br />
Next, students got in their groups and had a brief discussion about what they felt comfortable sharing. The discussion was reserved. While some students were happy to talk, I think many were feeling quite vulnerable and weren't entirely comfortable.<br />
<br />
<i>Word wall</i><br />
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On the whiteboard, I put a heading of "Thoughts/feelings", and encouraged students to write anything they wished relevant to this class. This was mildly terrifying. At first, only one or two students would slowly walk up. Then a few more, and then more. The atmosphere became markedly more relaxed. Students were seeing that they were not alone - others shared the same ideas, thoughts, and feelings, and I think it was comforting.</div>
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Then we spent a few minutes discussing what was up there. I summarized what I could, and students added a few more thoughts. Discussion was starting to roll.<br />
<br />
<i>Ideas</i><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrIIeIZ_wnzJzL93NujrAN-_S-BbZe6vr2vIU7NUiAYKdhDnj9PUE3fMMkZrnU3-K3Zh2DJ7-cCAnm4QTcbU0j3HmmWIp_KJ_mazLDjdD2XXtFem7C1CCNn312pjTpqts6Cmlj4woGK9s/s1600/20110922_124841.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrIIeIZ_wnzJzL93NujrAN-_S-BbZe6vr2vIU7NUiAYKdhDnj9PUE3fMMkZrnU3-K3Zh2DJ7-cCAnm4QTcbU0j3HmmWIp_KJ_mazLDjdD2XXtFem7C1CCNn312pjTpqts6Cmlj4woGK9s/s320/20110922_124841.jpg" width="320" /></a></div>
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Next, I had students brainstorm ideas for improving things. Class structure, assessments, practice, homework, behavior, teaching - anything and everything was fair game. Groups recorded their ideas, and then as a group, we transferred the ideas to the whiteboard up front. Once a few ideas were up there, more hands and ideas started popping up. I was impressed with the thought that students gave this.<br />
<br />
Next, I offered my thoughts on what I was seeing. I explained why a few things were non-negotiables, but left it open for discussion. For example, I do not review immediately before a test. I explained that this is because I try to base every decision that I make on two things: does it help learning, and does it promote accurate assessment. We don't review right before a test because it's not testing true learning - it's testing short-term memory. Not all students liked this, but most understood.<br />
<br />
Again, I was impressed with my students' engagement during this discussion. We were truly discussing learning and assessment. They were bring up things that they liked from previous teachers, and then we were discussing it in terms of "will this improve learning?" and "does this promote accurate assessment?" I've got a lot to think about here.<br />
<br />
<i>The Plan</i><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRsLz1fLUvVSyBb27iFmG1ZtRhkOjH-ldXjcCv7lIfZ8flALZ0o78rXEmZYTAi1tZIWoyY_OTMxClYnE0IlMsWWP7trthe1jPt801fSIcRfgJQjMYRaiIky6QTNCslAcv4vxq1wz_Rr8Q/s1600/20110922_125737.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRsLz1fLUvVSyBb27iFmG1ZtRhkOjH-ldXjcCv7lIfZ8flALZ0o78rXEmZYTAi1tZIWoyY_OTMxClYnE0IlMsWWP7trthe1jPt801fSIcRfgJQjMYRaiIky6QTNCslAcv4vxq1wz_Rr8Q/s320/20110922_125737.jpg" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEl_nXTqe2Djp5zDXkdV6duVdTtrEHBj5Qu4JTct3G5iyS0PYvbJuaKS8NgKcoVBEGengdllRPzzVlZPCQsra4g20bmKBcUHhPUa2G2wEBjBpW7sbnoiMc7WxrEyqoYvecVQqNmJ5sIWw/s1600/20110922_102810.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEl_nXTqe2Djp5zDXkdV6duVdTtrEHBj5Qu4JTct3G5iyS0PYvbJuaKS8NgKcoVBEGengdllRPzzVlZPCQsra4g20bmKBcUHhPUa2G2wEBjBpW7sbnoiMc7WxrEyqoYvecVQqNmJ5sIWw/s320/20110922_102810.jpg" width="320" /></a></div>
To wrap things up, I asked each group to come up with three specific, concrete things that we could change in the context of our discussion. Each group presented their three, and then we voted as a class what three things we could change. There was a remarkable similarity between the three blocks.<br />
<br />
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<div>
This was an incredibly intense day for me. I spent 4.5 hours discussing everything about how I teach. I am emotionally exhausted. It was incredibly powerful, but my energy is gone. </div>
<div>
<br /></div>
<div>
At the end of my third block, I was speaking to a few students in the last two minutes, and they asked me some innocent sounding questions that really capped off the day:</div>
<blockquote>
Why did you decide to teach? Are you sure you like teaching? You should have more fun!</blockquote>
I didn't have time to fully respond, and the questions were asked with true interest. I don't think they were suggesting I shouldn't teach. I think they were picking up on the fact that this job is hard, and this day especially was draining. In the 30 seconds I had, I said that I had been having this discussion for 4.5 hours today, and that it was scary for me. I was letting them write and say anything about my teaching, and I think they understood it.<br />
<br />
After the class, I took a walk around the school to let things soak in a bit. I'm tired. I don't know exactly what to do. But, I do know one thing. I need to show students my love. I do love this job, and I love my students. I don't think I share that. Enthusiasm and spirit are tough for me. I've always been a rather reserved, introspective person. People who get to know me learn to pick up my emotion and excitement, but I need to do more than that. But remain who I am.<br />
<br />
I need to figure out <i>my way</i> of showing students my love. For them, for my job, for my life.<br />
<blockquote>
</blockquote>
Unknownnoreply@blogger.com7tag:blogger.com,1999:blog-6708897041529904500.post-89554561522929995582011-09-06T21:59:00.003-05:002011-09-06T22:01:23.554-05:00When The Detour Has A Detour<div class="separator" style="clear: both; text-align: center;">
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I was driving to pick up my wife today, and the main road I needed was closed for construction. So I took the detour, and then came to a road completely closed off for an emergency. I then took my own detour.</div>
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As a new teacher, naturally all that I think about is teaching right now, and I thought this relevant. In the classroom, there will be detours. Sometimes they'll be predictable, sometimes they'll be surprises. The route may change, we may get lost, and maybe we'll see some things we didn't expect to. However, as long as I know where we are going, we'll get there.</div>
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I like that.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-72756061434590316912011-08-26T15:36:00.001-05:002011-08-26T15:36:38.968-05:00Doing MathematicsI had a thought this morning biking home from school (our first day is next Thursday). What is <u>the</u> goal I have for my students? Well, to learn mathematics. In order for that to happen, I need to<i> maximize the amount of time that students are actually doing mathematics</i>. Everything I plan should be centered around this.<br />
<br />
This begs the question, though - what is "doing mathematics"? First, it's easy to list a few things that it is not. Doing mathematics is not:<br />
<br />
<ul><li>taking notes</li>
<li>listening to me talk</li>
<li>drill & kill assignments</li>
<li>taking tests</li>
</ul><div>Already, so there is one guide post - minimize the amount of time doing these activities.</div><div><br />
</div><div>As for doing mathematics, I like the way that Keith Devlin describes it:</div><blockquote><span class="Apple-style-span" style="background-color: white; font-family: verdana, arial, helvetica, sans-serif; font-size: 13px;">"Doing math" involves all kinds of mental capacities: numerical reasoning, quantitative reasoning, linguistic reasoning, symbolic reasoning, spatial reasoning, logical reasoning, diagrammatic reasoning, reasoning about causality, the ability to handle abstractions, and maybe some others I have overlooked. And for success, all those need to be topped off with a dose of raw creativity and a desire - for some of us an inner need - to pursue the subject and do well at it.</span></blockquote><div>However, I'm having a difficult time describing exactly what "doing mathematics" means. In my mind, it's more of "I know it when I see it", so I need to work on clarifying what it means. Some activities that I think do fall under this umbrella:</div><div><ul><li>Solving novel problems</li>
<li>Student-generated questions</li>
<li>Investigations</li>
<li>Cooperative groupwork</li>
<li>Communicating reasoning and thinking</li>
<li>Proving conjectures</li>
<li>Finding patterns</li>
</ul><div>This list could go on and on, but my takeaway has been this. <i>All planning should be focused on maximizing the amount of time that my students are doing mathematics.</i></div></div>Unknownnoreply@blogger.com3tag:blogger.com,1999:blog-6708897041529904500.post-83081064368945632222011-08-18T16:01:00.001-05:002011-08-21T15:37:58.020-05:00Making Assessment Part of TeachingI spent the first part of the summer thinking about whether I should give standards-based grading a shot my first year or not. Actually, that's not true. I spent the first month of my summer in Germany and Spain. After that I started thinking about grading.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhazSaYTClp64UZQdvJi3h91w1-z6uFCJ6m382Lwd8MFIEyjmTIOJTLpgK58ACWnYwUx8QcsK7DVXxYqhd5dZ3NSr0orQqcXRFnrtY1JRS9L7qNz5XN3sfMENMSjm_yCQbgMw-l4nGHoxYs/s1600/IMG_3520.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhazSaYTClp64UZQdvJi3h91w1-z6uFCJ6m382Lwd8MFIEyjmTIOJTLpgK58ACWnYwUx8QcsK7DVXxYqhd5dZ3NSr0orQqcXRFnrtY1JRS9L7qNz5XN3sfMENMSjm_yCQbgMw-l4nGHoxYs/s320/IMG_3520.JPG" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">My wife and I hiking up to Schloss Neuschwanstein</td></tr>
</tbody></table><br />
Anyway, all it took was a little nudge from some teachers on Twitter along with support from my chair and administration. Over the past couple of weeks, I've been working on designing my grading system, and discovered a major reason to implement SBG: it actually gets me excited about assessment!<br />
<br />
During my student teaching, I felt that assessment was one of the worst parts of teaching. A necessary evil. I loved teaching, but when it came to giving tests and passing out only vaguely meaningful grades, it was just something I had to make it through. And then I had to deal with students negotiating for another point or pleading for extra credit. We were never discussing their learning, we were discussing how many points they had and how many they needed. Ugh.<br />
<br />
Now, when I'm working on designing my system and assessments, it feels more like teaching than test giving. The focus is specifically on what I want students to learn and their progress. Hopefully the conversations will mirror this.<br />
<br />
I think this alone is a good enough reason to try SBG.Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6708897041529904500.post-68331724932032219462011-07-18T21:00:00.001-05:002014-02-03T20:38:33.124-06:00The Coming Year - Technology In My Classroom<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
In a little more than a month, I will begin my first year as a high school math teacher. Having a vague understanding of how challenging it is going to be, as well as knowing my tendency to take on too much, one of my goals <i>before</i> the year starts is to have a plan for the technology I am going to use, its purpose, and how I am going to incorporate it into my teaching.</div>
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In choosing which technologies I will use, I am using the following as my requirements:</div>
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
</div>
<ul>
<li>Free to use as in cost. <a href="http://www.fsf.org/working-together/">Truly free software</a> is a bonus.</li>
<li>Cross-platform. Works with Windows, Mac, and Linux without a fight.</li>
<li>No downloads or installations required, other than a reasonably modern browser.</li>
<li>Limited number of registrations and sign ups.</li>
<li>Ease of use. I want to teach math, not technology.</li>
</ul>
<br />
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That being said, here are the weapons I plan on brandishing:</div>
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<b><a href="http://www.geogebra.org/cms/">GeoGebra</a></b></div>
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Absolutely amazing mathematics software designed for teaching and learning mathematics. While it's not always the most intuitive software to use, its dynamic and interactive nature make it a powerful tool in the classroom. There are loads of <a href="http://www.geogebra.org/cms/en/help">tutorials</a> and <a href="http://www.geogebra.org/en/wiki/index.php/English">ready-to-use materials</a> on the main website.</div>
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<b><a href="http://python.org/">Python</a></b></div>
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I absolutely agree with <a href="http://www.ted.com/talks/lang/eng/conrad_wolfram_teaching_kids_real_math_with_computers.html">Conrad Wolfram</a> that teaching a bit of programming is crucial in teaching mathematics in the twenty first century. Python is my programming language of choice for many reasons. Also, <a href="http://www.google.com/edu/computational-thinking/index.html">Google's Exploring Computational Thinking</a> is a great resource (and they even posted one of my lessons from my student teaching!).</div>
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<b><a href="https://docs.google.com/">Google Docs</a></b></div>
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If you have never used Google Docs before, I highly recommend it. I often hear it described as Microsoft Office in the cloud, but it is much more than that. It puts collaboration and sharing at the center of the process. Students are able to work on the same document at the same time, chat online together, track who made changes and when, and so much more. And it keeps getting better.</div>
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<b><a href="http://twitter.com/">Twitter</a>/<a href="http://plus.google.com/">Google+</a></b></div>
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For most of the summer, I simply assumed that I would be using Twitter, but the recent (beta) release of Google+ has been rethinking that. I love the idea of back-channeling, communication, and collaboration inside and outside of the classroom, as well as giving students a different way to contribute. Circles, hangouts, and sparks (Google+ features) are promising features. <a href="https://docs.google.com/present/edit?id=0AclS3lrlFkCIZGhuMnZjdjVfODk5aGhjZnJnZGc&hl=en_US">Here</a> is an evolving presentation on some potential uses of Google+ in education (and a great example of Google Docs in action!).</div>
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The math and tech geek in me is excited. These tools have the potential to engage students with mathematics at a level far beyond my high school experience, and that was not all that long ago.</div>
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Oh, and <i>please</i> leave comments! I would love to hear your thoughts and suggestions and learn about new resources.</div>
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<u><strike>Next post: How I plan on using technology for continual professional development and reflection.</strike></u></div>
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