tag:blogger.com,1999:blog-32485481978095932452024-03-05T05:43:52.714-06:00Trust Me - I'm a Math TeacherEducation, math, project-based learning, other stuff, and possibly ninjas.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.comBlogger38125tag:blogger.com,1999:blog-3248548197809593245.post-30373343792999929322014-08-28T12:55:00.000-05:002014-08-28T12:55:36.757-05:00A Brief Look at Phrasing Descriptions of FunctionsDuring a lesson this morning, students were asked to describe this function rule in their own words:<br />
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One student came up with this:</div>
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We had a brief group discussion about whether this phrase made sense or not. One student said that they might interpret this phrase differently:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp4U7JgMBvcwiOglSclAoWWczV3Ial6KWMysjozoMtr6OqNUfEy8Mxov9YkJTIRDWIMCCtAO9VXHXPTRJiBFbDlCRVr8DbjzVaW3L9Nm01PjJftdsKDMjtIjo6iK4nDe8hEWGKnQd3VxvL/s1600/Photo+Aug+28,+8+10+56+AM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp4U7JgMBvcwiOglSclAoWWczV3Ial6KWMysjozoMtr6OqNUfEy8Mxov9YkJTIRDWIMCCtAO9VXHXPTRJiBFbDlCRVr8DbjzVaW3L9Nm01PjJftdsKDMjtIjo6iK4nDe8hEWGKnQd3VxvL/s1600/Photo+Aug+28,+8+10+56+AM.jpg" height="320" width="240" /></a></div>
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Some students agreed, saying that the wording could possibly suggest that the sum was performed first, and then squared after. I asked them to think of a better way to say what was trying to be said, and they came up with this:<br />
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Not terribly different, but the order of the words made a lot of difference. The students agreed that this phrasing was clearer than the original.<br />
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Admittedly, this was a pretty simple problem, but the students brought up some good points in our conversation around it. It was a good opportunity to explore the nuances of being precise when talking about mathematics. Even one turn of phrase can be misleading; hopefully the students learned a bit about being careful about their wording while still being succinct.<br />
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<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com1tag:blogger.com,1999:blog-3248548197809593245.post-15486805840792776432014-08-27T14:56:00.000-05:002014-08-27T15:05:13.007-05:00First Day: Gathering Students' Impressions of MathIf there's one thing about teaching I'm not very great at (and there are many such things), it's the first day of school. I always struggle with it. I find myself so busy preparing for the year at large, or getting my classroom ready, or whatever else is demanding my attention, that I never really take the time to plan out a really great first day.<br />
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In part, I ended up doing what I described (in tongue-in-cheek fashion) to my students as the "time-honored tradition" of going over the syllabus for the first day of class. At one point, one of my administrators walked in to watch my class for a bit, and all they saw was me going over the syllabus. It was one of those <i>"please just kill me now"</i> moments for me.<br />
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I'm being over-dramatic, though. It really wasn't so bad. I'm really looking forward to working with the group of seniors I have this year, and I enjoyed meeting them today. I definitely did a lot of talking, which I never prefer to do, but it'll be different tomorrow.<br />
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As part of the first day of class, I had my students fill out a survey about how confident they feel about their math skills, what "doing math" means to them, and what they hope they'll have learned by the end of the course. The first three questions were Likert scale items. Here are some of the numbers:<br />
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<b>1. How confident are you in your ability to "do math"?</b><br />
<b><br /></b>
Completely confident: 11/78<br />
Mostly confident: 31/78<br />
Somewhat confident: 22/78<br />
A little confident: 6/78<br />
Not at all confident: 8/78<br />
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<b>2. How confident are you in your ability to talk about math verbally using mathematical reasoning and vocabulary?</b><br />
<b><br /></b>
Completely confident: 5/78<br />
Mostly confident: 16/78<br />
Somewhat confident: 31/78<br />
A little confident: 14/78<br />
Not at all confident: 12/78<br />
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<b>3. How confident are you in your ability to communicate about math in writing?</b><br />
<b><br /></b>
Completely confident: 7/78<br />
Mostly confident: 14/78<br />
Somewhat confident: 36/78<br />
A little confident: 18/78<br />
Not at all confident: 5/78<br />
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Overall, my students this year seem to be carrying a healthy level of confidence in their ability to "do math." (Of course, that depends on their definition of what it means to "do math," which I asked later on.)<br />
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There's a considerable split in confidence with my students as far as communicating mathematically. Those are two areas I intend to focus on this year: I want my students to speak and write confidently about mathematics. I want them to be well-versed in the <a href="http://www.corestandards.org/Math/Practice/" target="_blank">Math Practice Standards</a> by the end of the course.<br />
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There were some other short answer questions. There are too many responses to list, so I just picked a few examples that I think give the general view of the students:<br />
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<b>4. What do you think it means to "do math?"</b><br />
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"I think it means solving problems with numbers. Doing math is when you work out a math problem. Also taking time to make sure your answer is right."<br />
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"To do math is understanding the logic behind a problem. It is the ability to explain problems to others verbally and on paper. Doing math is using more than one technique to find the correct answers."<br />
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"I believe that 'doing math' is thinking about a problem critically and using certain formulas to find out the answer to something."<br />
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"To do math is to find the answer to a problem that involves numbers, distances, functions, or any form of measurement. A math problem usually has a set number of answers that have to be found through use of mathematical functions or equations. But to do math is to use logic to solve something."<br />
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"Doing math means to completely understand it, and for me that comes in 3 parts. Before you can properly plug in numbers to equations, you must first know what those equations mean, and what answer(s) they are trying to achieve. After knowing that, you must know how to correctly plug in the numbers in the equation to get your answer. The final thing that you need to know how to do when 'doing math' is being able to explain what you did, and why. If you are not able to explain how or why you did what you have done, then there is no way to tell if you were right in your thinking."<br />
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"'Do math' to me means to solve a puzzle. You need to find all the pieces of the puzzle in order to solve the problem."<br />
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"To 'do math' is to have an answer to the problem presented. However, I think that 'doing math' also includes the full understanding of the problem. Also being confident in the answer that you have."<br />
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<b>5. What does it mean to be a "good mathematician?"</b><br />
<b><br /></b>
"Math is easy to learn but hard to master. Given enough time, anyone can solve any problem. Being a good mathematician means being able to solve equations in a quick manner."<br />
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"Being a good mathematician means that you can easily identify and solve problems quickly and correctly."<br />
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"A good mathematician doesn't give up easily, but keeps trying different methods until the problem can be solved. A good mathematician learns to apply conclusions to the world surrounding him or her."<br />
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"A good mathematician is not necessarily someone that finds answers quickly, but rather one that finds answers effectively."<br />
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"A good mathematician is someone who can answer the problem that they have set in front of him or her. They can execute the best possible method of doing a problem, in the quickest way possible. They also understand all of the math behind it."<br />
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"A good mathematician would use... nothing other than your brain. Wouldn't use a calculator and know every function in math. Be like Albert Einstein."<br />
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"Being good at math means being able to remember formulas and solve problems quickly. I also think it means being able to help anyone when they need help during a certain area they don't quite understand."<br />
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"To be a good mathematician means you have a brain like a computer. If someone asks you a difficult math question you should be able to answer it in a matter of seconds."<br />
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"A good mathematician would know how to recognize a math problem. A good mathematician would actively seek answers to things he/she doesn't understand. Finally a good mathematician knows and studies deeply the subject of math."<br />
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<b>6. What do you hope you will have LEARNED in Pre-Calculus by the end of the school year?</b><br />
<b><br /></b>
"I want to learn how to solve math problems in the quickest ways possible. I also want to explore different forms of calculators and their functions."<br />
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"I hope at the end of the year I learn how to solve my problems, without errors or depending on anyone for help."<br />
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"Pre-Calculus should teach students more advanced forms of mathematics, past the formulas and equations. Pre-Calc is a dreaded class by some, but can be helpful in certain professions."<br />
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"I really want to know how some advanced math could be used to solve everyday problems, so if it is just the same old stuff revisited from last year at least show us how it applies to real life."<br />
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"I honestly just hope to learn something new in Pre-Calculus. I want more challenging problems so I can have more math skills."<br />
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"A way to understand Calculus without being a mindless zombie to the textbook. Well, understand enough to understand college Calculus."<br />
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"I hope that I have learned new formulas and learned them well."<br />
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"Hopefully I will be able to pass."<br />
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"I hope that I will have learned to explain my reasoning with most of my math problems, thus broadening my horizon on how to be a good teacher."<br />
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<b>7. What do you hope you will have EXPERIENCED in Pre-Calculus by the end of the school year?</b><br />
<b><br /></b>
"I hope to experience an even greater understanding of math as well as enjoy it more. It's currently my favorite subject, so I believe that most, if not all, of my experiences will be positive in this class."<br />
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"I hope that I will experience how to speak mathematics in a different kind of language than what I usually use when I explain a solution to a problem."<br />
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"Uhm, what am I SUPPOSED to have experienced? I don't really have any hope for anything in this class."<br />
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"By the end of the year I hope to have experienced how to deal with stress when it comes to math. Math has always been my worst subject and I get stressed a lot while doing math."<br />
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"I hope to experience new things and different ways of solving problems."<br />
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"I really don't know. Surprise me."<br />
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"I hope to experience what it will be like to use math in the real world, such as: taxes, sales, etc."<br />
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"I hope to have experienced the questions that make you sweat, and look back in your notes to figure out. I love puzzles and math and I love a challenge so I want to experience a good challenge in a math course. I want to be able to help others with their homework and also be able to say I had the best Pre-Calc teacher in high school history." <i>(Geez, no pressure there, right?)</i><br />
<i><br /></i>
<i><br /></i>
While I definitely don't think this first day of school was the greatest, I <i>did</i> end up getting a lot of really thoughtful responses to these questions (again, way too many to list). The attitudes and views of my students towards math definitely cover a wide spectrum this year. I'm really encouraged by the number of students who said they're <i>craving challenge</i>. I love it. I hope I can deliver.<br />
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We're starting a group task by the end of the week. I'm going to try grouping students so that each group member has a certain level of confidence in talking about math, writing about math, or just doing math. I may also group them by how they responded to the written questions. We'll see how it goes.<br />
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It will be interesting to see how the students answer these questions in May. I hope that more of them will see "doing math" in terms of problem-solving, constructing arguments, modeling, looking for structure, and so on.<br />
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And so a new school year begins. <i><a href="https://www.youtube.com/watch?v=3IlJFNMAJ-k" target="_blank">Allons-y!</a></i>Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com8tag:blogger.com,1999:blog-3248548197809593245.post-62750786459251957232013-12-20T13:33:00.001-06:002013-12-20T13:33:33.848-06:00Leaving It All On The FieldAs far as the school year goes, winter break is halftime. I'm exhausted. I feel like I'm going into the locker room having left it all on the field.<br />
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After what felt like 12,383,908,786,358,213 years (give or take a few), the long wait for winter break is finally over. In a few minutes, the dismissal bell will ring and we'll all be running out the door for a glorious two weeks away from work.<br />
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I've been feeling that really prolonged kind of tired these past couple of weeks. The kind of tired that teachers feel after a few solid months of establishing classroom routines, figuring out the best ways to help students learn, continually assessing & giving feedback, communicating with parents, collaborating with other teachers, and getting involved with other facets of the school community behind the scenes.<br />
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It's been a crazy-busy first semester for me, and I haven't had time to blog about all of it. I'm on a district-level committee working on developing a new evaluation tool for the teachers in our district, necessary because of changes made to our state's laws. I've given up most of my lunches for Anime Club and to help students with re-taking quizzes, catching up on missing work, or getting extra help. For the past month, I've also been involved in a secret project (shh!) that I'm pretty excited to be a part of. I also had the pleasure of bringing my old math ed professor from MSU into my classroom to check out what I've been doing; and in turn, I had the pleasure of being on a guest teacher panel for his class of up-and-coming teacher candidates. We had great conversations about how to reach students and build relationships, and it was great to see him in action with a few of my own students. I'm still learning from him.<br />
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As the calendar year winds down, I find myself thankful for a break. I'm looking forward to spending time with family and friends, going home for Christmas, celebrating the new year with my wife, and getting the chance to relax and re-energize.<br />
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To all of my teacher friends, colleagues, and acquaintances, I wish all of you a Merry Christmas and a Happy New Year!<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com2tag:blogger.com,1999:blog-3248548197809593245.post-82478964065862325992013-11-12T14:24:00.001-06:002013-11-12T14:24:06.461-06:00Twosday Things: Ingenious Responses. Also Fish.Time again for Twosday Things!<br />
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<b>Thing #1:</b><br />
The other day, I stepped out of my classroom for a moment. When I came back, one of my students had drawn this on the board:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFES1Hxvr9dP0m_QKyzC4dEqME18drEBPX_fRv1tOw4gvYC7wV9xX1bfyaRrt9B_j5RCsZ6Kw-FFfIFlqTmNTsIRj4jjq3Qkwc-t3uJWzfu8_roGDCeSWvJCaEuCKbad-BvSbIv8wot8WO/s1600/BYzQRe_CAAAXrQ6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFES1Hxvr9dP0m_QKyzC4dEqME18drEBPX_fRv1tOw4gvYC7wV9xX1bfyaRrt9B_j5RCsZ6Kw-FFfIFlqTmNTsIRj4jjq3Qkwc-t3uJWzfu8_roGDCeSWvJCaEuCKbad-BvSbIv8wot8WO/s400/BYzQRe_CAAAXrQ6.jpg" width="400" /></a></div>
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I took one look and figured, "what the hell, I'll tweet it." So I did:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTd_uqnNBU39toeRCPSAzIOVl-pZenykb5jX2Hm_r3Ogy_STtAAtC3BIuyode4dH3yRmeaJNlcGs0TPhyRCEevGuPvXj7t22a3Bskn60zVcL1V8uLXuiP2-ZjhsyRNiU8IvK_hoHKkl__4/s1600/Screen+Shot+2013-11-12+at+1.21.52+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTd_uqnNBU39toeRCPSAzIOVl-pZenykb5jX2Hm_r3Ogy_STtAAtC3BIuyode4dH3yRmeaJNlcGs0TPhyRCEevGuPvXj7t22a3Bskn60zVcL1V8uLXuiP2-ZjhsyRNiU8IvK_hoHKkl__4/s1600/Screen+Shot+2013-11-12+at+1.21.52+PM.png" /></a></div>
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One reply stated that this was probably a <a href="http://www.youtube.com/watch?v=Rvtq7C2JFAU" target="_blank">reference to Fairly Oddparents</a>, which given the age of my current students wouldn't surprise me.<br />
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However, the prize for Most Brilliantly Mathematical Response definitely went to <a href="http://mathiex.blogspot.ca/" target="_blank">Gregory Taylor</a> (<a href="https://twitter.com/mathtans" target="_blank">@mathtans</a> on Twitter):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl5SmsLF4-prac74vjJaylHV98dNYbPUzlRZeMQcPNEIwU4RgfMEfgHnoSLp7aFmiBQxsg3JLK0ykr9hSRRedhX46mEPblOeixuake1jND044SjdySApeDOK344adVGgF8kYp_fsdEQpS7/s1600/Screen+Shot+2013-11-12+at+1.24.19+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl5SmsLF4-prac74vjJaylHV98dNYbPUzlRZeMQcPNEIwU4RgfMEfgHnoSLp7aFmiBQxsg3JLK0ykr9hSRRedhX46mEPblOeixuake1jND044SjdySApeDOK344adVGgF8kYp_fsdEQpS7/s1600/Screen+Shot+2013-11-12+at+1.24.19+PM.png" /></a></div>
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I feel like if I'd gotten that kind of response from a student, I'd have just given them an A for the semester right then and there. (Okay, maybe not. But I'd be impressed.)<br />
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<b><br /></b>
<b>Thing #2:</b><br />
One thing I've noticed about my teaching practice this year is that I've become more open-minded with how students respond to questions and problems.<br />
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Here's an example of what I mean. One of my students came to me today with the following solution to a problem:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnqQUXYnuXc_GYd8j1J04Y9Z0FBzrBNn1wAtOID2txP4-RoFDVe4VpWutlepeobx_G6g0x6WgFkumV6caCu9x_UaOrxj4DObYR5yoNaKYmjz1Wd4mQa_tNisyyg5TyWJS6xXu9vuzIdy-B/s1600/Photo11121004.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnqQUXYnuXc_GYd8j1J04Y9Z0FBzrBNn1wAtOID2txP4-RoFDVe4VpWutlepeobx_G6g0x6WgFkumV6caCu9x_UaOrxj4DObYR5yoNaKYmjz1Wd4mQa_tNisyyg5TyWJS6xXu9vuzIdy-B/s640/Photo11121004.jpg" width="640" /></a></div>
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Two disclaimers: (1) The student obviously took some "mathematical liberties" when drawing this diagram. (2) The student did much of their work without a calculator, but explained to me in person what was done: he used the distance formula to calculate the length of each side, then used the Pythagorean Theorem to see whether the three sides formed the sides of a right triangle.<br />
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Out of context, this seems like a perfectly reasonable way to solve to problem.<br />
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However, this actually came from a problem set focused on parallel and perpendicular lines. The solution path I was "looking for" was to calculate the slope between each pair of vertices and determine if there were two sides that were perpendicular to each other.<br />
<br />
What's my point here?<br />
<br />
A year or two ago, this is probably how I would have responded to the student's work: <i>"Um... well, that's ONE way to solve it I guess, but I was really looking for [insert what I was looking for]."</i><br />
<i><br /></i>
But today, this is how I responded: <i>"Whoa, that's brilliant! I hadn't actually thought of solving the problem that way, but that makes a lot of sense! This is genius!"</i> And I followed that up with an explanation of how most other students were solving the problem by calculating slopes as I described above; but the student's mathematical reasoning was both valid and awesome.<br />
<br />
This is a great example of how I've changed as a teacher this year. I've always been okay with students coming up with different solution paths to problems; however, I often tried to steer them toward <i>particular</i> solution paths, even if what my students were doing was perfectly reasonable.<br />
<br />
Insisting on particular solutions paths isn't, in and of itself, a bad thing. There are situations where it's good to train students on solving a problem a particular way; doing so adds to their "mathematical toolbox," equipping them with a variety of skills for solving problems.<br />
<br />
But there are times, I think, when we as math teachers need to be okay with students solving problems in unexpected ways. I think this instance was one of those times. This was a student who had been struggling with math at times this year, but today he came to me with a brilliant solution that I wasn't expecting to see. <i>That</i> deserved praise and recognition.<br />
<br />
As I said, a year or two ago, I would have been "just okay" with the method my student used to solve the problem, but not all that enthusiastic because he hadn't done it the way I was trying to teach.<br />
<br />
I shudder to think that, just a year or two ago, I wouldn't have embraced his work as enthusiastically as I did today. If I had responded with, "<i>Well, that's one way to do it, but..."</i>, I probably would have done harm to the student's mathematical confidence. He applied previously-learned mathematical knowledge to a different type of problem. How could I have any problem with that?Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com4tag:blogger.com,1999:blog-3248548197809593245.post-84566833091690223442013-11-05T14:09:00.002-06:002013-11-05T14:11:40.804-06:00Twosday Things: Solution Methods and Student InitiativeIt's Tuesday, which means it's once again time for Twosday Things!<br />
<br />
In my dauntless endeavor to blog regularly, I am continuing to write about two (big or small, mostly small) things that happened in my teaching world over the previous week. This makes the third week in a row. Not bad.<br />
<br />
Before you read on, be sure to open up <a href="https://twitter.com/emergentmath" target="_blank">Geoff Krall's</a> awesome <a href="http://emergentmath.com/2013/10/30/a-problem-based-learning-starter-kit/" target="_blank">PrBL starter kit</a> in a new tab; this should be your next bit of reading after you're done here. You're welcome!<br />
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<br />
<b>Thing #1:</b><br />
One of my students (we'll call her Susie) was having trouble working through the following problem:<br />
<br />
<i>"Find the value of two numbers if half the larger number plus two equals the smaller number and their sum is 44."</i><br />
<i><br /></i>
Setting up a system of equations to represent the problem wasn't terribly much of an issue; Susie was able to do this on her own with a bit of questioning from me to prompt her thinking.<br />
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After we had set up the system, Susie seemed stuck on what to do next (though we had been working on systems of equations all week and I'd seen her succeed in completing similar problems).<br />
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Before I even said anything, another student (let's call her Nadia) offered to help explain what to do next, and I gladly obliged. Nadia used the <a href="http://www.purplemath.com/modules/systlin5.htm" target="_blank">elimination method</a> to solve the problem while explaining her steps to Susie:<br />
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After Nadia finished, Susie seemed confused. She understood that 28 and 16 had to be the numbers described in the problem, but she wasn't clear on the elimination method that Nadia had used. "I actually thought you were supposed to plug the equation for <i>b</i> into the second equation," she said. Susie proceeded to solve the problem using the <a href="http://www.purplemath.com/modules/systlin4.htm" target="_blank">substitution method</a>:<br />
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I found what happened next to be interesting: Nadia seemed confused about the method that Susie had used to get her answer, even though they both came up with the same thing! I said to them, "so Susie, it sounds like you were confused when Nadia used elimination to solve this problem, and Nadia, it sounds like you were confused when Susie used substitution." We talked about it, and both girls said the methods they each used just made more sense to them. I stressed to them that it was important to understand <i>both</i> of these methods (as well as solving by graphing), but also that it was great to see that each of them had their own way of figuring out this problem. <a href="http://brennemath.blogspot.com/2013/10/multiple-solutions-follow-up-to-when-is.html" target="_blank">As has happened in my class before</a>, students are seeing there can be more than one path to a solution.<br />
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<br />
<b>Thing #2:</b><br />
The above situation touches on something that has been developing among my students in my classes over the past few weeks: they're starting to regularly help each other out on their own.<br />
<br />
Stuff like the above has been happening with greater regularity in my classes. It's happened faster among my advanced students, but my other students are starting to do it as well.<br />
<br />
A lot of classroom time is spent allowing the students to work through problems at their own pace, with me providing one-on-one or small-group assistance as needed. This can be challenging to manage, particularly with having my "regular" and "advanced" classes in my room at the same time every period.<br />
<br />
That's part of the reason why I love it when students start to take the initiative and help each other out. I also love this kind of initiative because it's so important for students to be able to take agency of their own learning. It's an important life skill (at least I think so).<br />
<br />
One of my classes has really figured this out. Every day, they come in and they all get with their usual groups (I did no grouping; they formed these groups on their own and they work really well). They figure out what their assignments are. If I don't have a new workshop or learning module for them, they get to work and ask me for help whenever they have questions. They're also getting very good at helping each other out, checking each other's work, and asking each other questions (Thing #1, above, happened with this group).<br />
<br />
When students figure out how to take charge of their own learning, good things happen. One student (let's dub this one Marie) had been struggling all year with math. Last week, the small group of friends Marie regularly works with really focused on helping her understand how to solve systems of equations. They were able to give her a greater amount of attention and assistance than I was able to by myself. After a while, Marie started to be able to solve systems of equations on her own; she even got so excited about getting a problem right, that she wanted to do it on the board! AWESOME!<br />
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It's not like this every day, and it's not like this in every class. But it's starting to happen more, and it's great to have one class that's really taken off with helping each other out.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com2tag:blogger.com,1999:blog-3248548197809593245.post-11522542931866767872013-10-29T13:31:00.001-05:002013-10-29T13:31:25.410-05:00Twosday Things: Hearts, Stars, Messy NumbersTime again for Twosday Things!<br />
<br />
Taking a cue from <a href="http://brennemath.blogspot.com/2013/10/two-things-from-tuesday.html" target="_blank">last Tuesday's post</a>, I'll discuss two teaching-related things (however big or small) that happened over the past week. I'm trying to post about two things every Tuesday throughout the school year (hence the title, "Twosday Things"). This makes the second week in a row; so far, so good.<br />
<b><br /></b>
<b>Thing #1:</b><br />
Something I've noticed that happens A LOT in my class:<br />
<br />
<ul>
<li>Student is working through a (typically algebraic) problem.</li>
<li>Student gets a non-integer answer (i.e. a "decimal answer").</li>
<li>Student immediately assumes they must be wrong. Often accompanied by asking the teacher, "am I <i>supposed</i> to get a decimal for my answer?"</li>
</ul>
<br />
This is a near-daily occurrence in my class, despite my frequent insistence that "decimals are numbers, too!" ("Fractions are numbers, too!" is similarly used often.) I cannot even count the number of times this happens in a school year.<br />
<br />
How does this happen? How do our students reach the point where they automatically assume that "decimal answers" must be wrong? How do we let them get to high school with this assumption cemented into their mathematical psyche?<br />
<br />
Yesterday, I took this question to my <a href="https://twitter.com/brennemania" target="_blank">Twitter feed</a>:<br />
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Some super-awesome math-types from the Twittersphere chimed in with their thoughts on the topic:<br />
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<br />
"Give them messiness." I love that. I feel like our students need more practice and earlier exposure to "messy numbers," because real-world math is messy and complex. Students need to learn that decimals, fractions, irrationals, etc. are all numbers, too.<br />
<br />
At the same time, I don't think it's <i>inherently</i> bad that students question their answers every time they get something "messy." Sometimes (often, in fact), their answer actually <i>is </i>the result of a mathematical mistake, and they need to be able to figure out where the mistake was made.<br />
<br />
I can see some potentially good habits here: stopping to think about whether the answer makes sense in the context of the problem; double-checking work for mathematical mistakes; and so forth. I just don't think that "getting a messy answer" should be the sole reason a student thinks they did something wrong. If anything, students should be trained to question "messy" answers <i>and</i> "clean" answers. Students should be in the habit of doubling back and re-checking their work to make sure their reasoning makes sense.<br />
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Maybe the mistrust in "messy" numbers can be a good thing; but if it is, it needs to be applied to <i>all</i> numbers. Equal opportunity, darn it!<br />
<br />
<b><br /></b>
<b>Thing #2:</b><br />
Today in class, I had a few students who asked for help with the following problem:<br />
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We discussed the fact that the problem mentioned "two numbers." We had no idea what those two numbers were, offhand. But, we had enough information to be able to set up a couple of equations. We just needed to pick two variables to represent the numbers first.<br />
<br />
"We can call these two numbers anything we want," I said. "We can call them <i>x</i> and <i>y</i>. We can call them <i>a</i> and <i>b</i>, or <i>c</i> and <i>d</i>. We could even call them stuff like, 'dollar sign' and smiley face.' What do you want to call these two numbers?"<br />
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One of my students said, "heart and star."<br />
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Math, learning, and hilarity ensued:<br />
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I had a terrible time keeping a straight face, especially when I said things like, "so what expression do we plug in for heart?" or "yep, we have to simplify by combining our star terms, so star plus eight equals twenty-four," or "there we go, star equals sixteen and heart equals forty."<br />
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It was a fun little way to talk about the concept of representing unknown values with variables. Why settle for boring old <i>x</i> and <i>y</i> when you can have a bit of fun?Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com4tag:blogger.com,1999:blog-3248548197809593245.post-30902514283164927112013-10-25T13:36:00.000-05:002013-10-25T13:36:00.064-05:00For Posterity: An Awesome Teaching DayThere are about 180 days in a school year. Multiply that by the number of years you've been teaching. That's a crap-ton of days.<br />
<br />
Many of them are good, or just okay. Many of them leave you wondering whether or not you're any good at teaching at all. A few of them turn out to be terrible.<br />
<br />
But once in a while, you have a day that makes living through all of those other days <i>completely worth it. </i>For me, yesterday was one of those days.<br />
<br />
Last night, I was recognized by our district's Board of Education for "contributions to the school and the community." It was a pretty cool honor.<br />
<br />
I was already having an awesome day. I finished grading quizzes that my students had taken this week. They did well overall, but I was particularly impressed by many of my students who had been struggling. Those students have been making a point of coming to me for help, participating more in class, and generally just turning things around. They've been busting their butts, and their work is paying off. It's thrilling to see those students experience success in my class; there have been smiles and high fives all around.<br />
<br />
In the evening, I attended the Board meeting to be recognized along with a few other students and staff from our district. When my name was called, I went up to the front to receive my framed certificate:<br />
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After my principal said a few things about me, I had a moment to shake the hands of every Board member.<br />
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One of them, whose son graduated a couple of years ago (and whom I had taught for two years) stopped me for a moment to tell me: "I want you to know, my son has decided to switch his major to math because of you."<br />
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Wow. I mean, just wow. That pretty much describes what I felt at that moment. I didn't really know what to say. I think I almost started crying, or at least got a bit teary. It was just so awesome to hear those words.<br />
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That news, that moment, meant a hundred times more to me than any award or certificate. The feels. <i>The feels.</i><br />
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As a teacher, when you have students, you don't always know whether or not you've having any kind of impact on their lives. Most of the time, I feel like I have no idea whether or not I'm having any impact. I guess sometimes you don't find out for sure until after those students graduate and move on. But however long it takes, when you <i>do</i> find out you had a positive impact on a student, it's completely worth it. Nothing else really matters.<br />
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Somewhere down the line, I'm going to have a terrible day and I'll need something to lift my spirits; that's why I had to write about my awesome day. I'll always have this to come back to when I need to be reminded why teaching is so rewarding and worth everything.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com1tag:blogger.com,1999:blog-3248548197809593245.post-27577521058316982522013-10-22T14:29:00.000-05:002013-10-22T14:29:11.247-05:00Two Things From a TuesdayOr maybe I should title this post "Twosday Things." Because I like portmanteaus.<br />
<br />
<b>Thing #1:</b><br />
Today, I was talking one-on-one with a student about functions. We were talking about the relationship between domain and range, and how to tell if two sets of values make up the domain and range of a function. We talked about how values in the domain are each assigned to one and only one value in the range by the function. I chimed in with the "mailbox analogy" to further explain the relationship: say you're mailing a bunch of letters. The stack of letters is like the domain, and the houses the letters are being mailed to are like the range. You can mail multiple letters to the same house, but you can't mail the same letter to multiple houses. "So you can't mail the same letter to Chicago, New York, and San Francisco simultaneously," I said to the student.<br />
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"Unless it's e-mail," the student replied.<br />
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<i>HOLY CRAP.</i> That was a really, <i>really</i> good point! I was utterly stunned that I hadn't thought of that. I guess the analogy kind of breaks down in that regard if you throw e-mail into the mix. I'm still pretty sure I got my point across, but it does have me thinking about the analogy I'm using to describe how functions work. Will this be an outdated analogy in the near future?<br />
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Either way, I was super-impressed by my student today.<br />
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<b>Thing #2:</b><br />
Some of my students are currently working on compound inequalities. Below is a piece of student work that I found interesting:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjw69j5w_uP9Um9T4bBlrWXZDOY46UszdzI2_Kf8Zs2p3zxkp-zQCEve_zzOeS8ks2PTp7jK_mjjODt0D2DXaKysiaRpEYI50alYRFhyphenhyphenFD8EgB4AHJuuRDj1enCAeF9I_0sxjSVBjJ0terf/s1600/Photo+Oct+22%252C+8+21+06+AM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjw69j5w_uP9Um9T4bBlrWXZDOY46UszdzI2_Kf8Zs2p3zxkp-zQCEve_zzOeS8ks2PTp7jK_mjjODt0D2DXaKysiaRpEYI50alYRFhyphenhyphenFD8EgB4AHJuuRDj1enCAeF9I_0sxjSVBjJ0terf/s400/Photo+Oct+22%252C+8+21+06+AM.jpg" width="400" /></a></div>
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The left side of the compound inequality vanished! I've actually been seeing this happen with several students in my class; every time they get one side of a compound inequality equal to zero, they omit it in the rest of their work.<br />
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I've been wondering where this is coming from. I imagine it might have something to do with the fact that students are sometimes taught about the existence of an "implied" zero that isn't actually shown. (For example, what is the slope of the line <i>y </i>= 2? There's no <i>x-</i>term, but there's an implied "0<i>x</i>" in the equation; thus, <i>y</i> = 0<i>x</i> + 2, and the line has a slope of 0.)<br />
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Maybe it's coming from somewhere else. I don't think it's anything <i>I've</i> done, but I could be wrong.<br />
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Anyway, that's two things from a Tuesday. Maybe I'll try to do this weekly, so I'm blogging more often.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com9tag:blogger.com,1999:blog-3248548197809593245.post-72993090336902824622013-10-08T22:01:00.001-05:002013-10-09T08:12:18.481-05:00Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")A couple of weeks ago, I wrote <a href="http://brennemath.blogspot.com/2013/09/when-is-right-answer-right-answer.html" target="_blank">this post</a> about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)<br />
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I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.<br />
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Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEir0AZDozisBxf7kuRdLvQirtadSHZtezUDyToXrLDW90WOaA7sxTGb5MC7eBTKPXZHVNTkb1hUKqV8Bigpx1qgjP-X1CaojdE5ECg1YzId3X5vA2y7HF8-3hricP3_QpT5JhW-SjmjuNdE/s1600/Photo+Oct+07,+12+30+30+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEir0AZDozisBxf7kuRdLvQirtadSHZtezUDyToXrLDW90WOaA7sxTGb5MC7eBTKPXZHVNTkb1hUKqV8Bigpx1qgjP-X1CaojdE5ECg1YzId3X5vA2y7HF8-3hricP3_QpT5JhW-SjmjuNdE/s640/Photo+Oct+07,+12+30+30+PM.jpg" width="640" /></a></div>
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<span id="goog_1662575099"></span><span id="goog_1662575100"></span><br />
So both students used point-slope form for their equations, and came up with two answers that <i>looked</i> different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.<br />
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I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're <i>both</i> right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were <i>both</i> right. It was my favorite moment of class from yesterday.<br />
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Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.<br />
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One student was adamant that the "first" point, (-4, 3), <i>had</i> to be plugged in for point-slope form instead of the "second" point, "because they're X<sub>1 </sub>and Y<sub>1</sub>," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:<br />
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And point-slope form was written on the board as Y - Y<sub>1</sub> = (X - X<sub>1</sub>). So I could see where she was coming from.<br />
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I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X<sub>1</sub>, Y<sub>1</sub>) and (-4, 3) as (X<sub>2</sub>, Y<sub>2</sub>).<br />
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My next question was, "So would that change things?<i> </i>Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.<br />
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I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.<br />
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We decided to have each of them solve their equations for <i>y</i>, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."<br />
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I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com0tag:blogger.com,1999:blog-3248548197809593245.post-56402168039229044632013-10-07T12:22:00.000-05:002013-10-08T16:02:46.096-05:00Taking a Teaching MulliganSometimes, despite trying to do my best job possible as a teacher, I screw up. I'm pretty sure it's healthy to accept that it happens from time to time.<br />
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A few weeks ago, my students took a quiz that pretty much nobody did well on. Like, not even really that close. (I'm not going to go into what the subject matter was or how my lessons were designed or what scaffolding I did -- that doesn't really pertain to the message of my post today.)<br />
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Needless to say, this elicited an emotional reaction from me. I actually had to stop grading and walk away for a few minutes because I was feeling a mix of sadness and anger all at once. I reasonably sure that I was uttering curse words under my breath after I came back and continued grading.<br />
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I think, unfortunately, there are some teachers who probably would have taken that anger and directed it at their students the next class period. I've seen teachers get absolutely pissed off at their students for doing terribly on a quiz or a test as a whole group.<br />
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I'm not one of those teachers. When students don't perform well on an assessment, I blame myself. I blame myself pretty hard, actually. Maybe more than I should. I guess I can't help it.<br />
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This happened on a Friday afternoon. I thought about what to do all weekend. I came back to my students on Monday and, in each class, just laid it out for them:<br />
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"Guys, nobody did well on this quiz. I'm sorry. I blame myself for that. When nobody does well, that tells me that I probably did something wrong with my teaching. So, I'm not going to include these quizzes in your grade for now. We'll come back to it next week, I'll try to teach differently, and we'll re-take this quiz. Does that sound fair?"<br />
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And it sounded fair to everyone.<br />
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I imagine part of why my students were amenable to this is because many of them sensed that they hadn't done well. I bet many of <i>them</i> were afraid they'd let <i>me</i> down, or that <i>I</i> was going to be mad at <i>them</i> for failing one silly math quiz. They probably don't know that, when a class bombs an assessment, the first question I always ask myself is, "what did <i>I</i> do wrong?"<br />
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Stuff like this is a humbling reminder that, even though I work hard and try my very best as an educator, there will be times where I come up short. I try to keep those instances few and far between, but from time to time it will happen. When it does, I think the right thing is to give my students a second chance -- or, more accurately, ask my students to give <i>me</i> a second chance.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com2tag:blogger.com,1999:blog-3248548197809593245.post-6713142250847270202013-09-29T11:54:00.003-05:002013-09-29T11:56:17.027-05:00When Is the Right Answer the Right Answer?This week, my students have been working on determining equations of a line based on properties of parallel and perpendicular lines (GRE 604 from the <a href="http://www.act.org/standard/planact/math/" target="_blank">ACT College Readiness Standards for Mathematics</a>), which involves problems like this one:<br />
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Several concepts popped up throughout the week while working on this skill: determining slope, slope-intercept form, point-slope form, and the relationships of slopes between lines that are either parallel or perpendicular to each other.<br />
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Throughout the week, I have been insisting that my students give their solutions to these problems in slope-intercept form, as shown in this student's work:<br />
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<span style="font-size: xx-small;">(There are some other things going on here that would also be interesting to talk about, but that will have to wait for another day.)</span> </div>
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Perfectly reasonable solution method, isn't it? Put the original line equation in slope-intercept form, determine the slope, use point-slope form to get the equation of the parallel line, and then solve for <i>y</i> to put that equation in slope-intercept form.<br />
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This morning, I found myself wondering <i>why</i> I was insisting on having my students put their answer in slope-intercept form.<br />
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Is it really necessary? I mean, couldn't the student have just stopped at point-slope form and still been correct? I mean, plug a few things into <a href="https://www.desmos.com/calculator" target="_blank">Desmos</a> and it's hard to argue otherwise:<br />
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I've been thinking about this and struggling with this all morning. The focus of this particular ACT skill isn't necessarily for students to determine the equation of line and put it in slope-intercept form; the skill is just to determine the equation of a line based on properties of parallel and perpendicular lines.<br />
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In the problem above, the student is given the equation of a line and a point on another line that is parallel. The student knew to look for the slope of the original line, knowing that the parallel line they were looking for would have the same slope. After determining the slope, the student created the equation of the parallel line using point-slope form.<br />
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Should it stop there? After all, the student correctly applied the properties of parallel lines and determined a correct equation. That's what the skill is all about, right? Why was I insisting that the student put their answer in slope-intercept form? I'm not sure it's necessary, and I think it also creates a situation where the student can make a simple algebra mistake and come up with an equation that is no longer "correct." On the other hand, expecting students to be able to put the equation in slope-intercept form isn't all <i>that </i>unreasonable, is it? After all, the student did just that with the equation of the original line in the problem, in order to determine the slope of the parallel line. Is that a good enough reason to insist on it, though?<br />
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This is just one specific case. I know this isn't the only instance in mathematics where something like this happens. When is the right answer the right answer?Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com6tag:blogger.com,1999:blog-3248548197809593245.post-42917391924179380802013-09-18T13:09:00.001-05:002013-09-18T13:09:22.282-05:00Coffee Spills! Sales Sheets! Math!I like to think I have good taste in music. When I was a kid, I played a lot of video games. Video games are super fun. The best part about video games, arguably, is the music. I will always hold the opinion that the Super Nintendo era gave us some of the best video game tunes in the history of ever. EVER.<br />
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So, these days I listen to a lot of video game music (VGM) cover bands. One of my current favorites is a recently-formed act, <a href="http://the-returners.com/" target="_blank">The Returners</a>. They're based in Austin, TX and they totally rock.<br />
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But they don't totally rock just because of their music. They totally rock because the band's founder, Lauren Liebowitz, recently helped me out with putting together a math task that involved coffee spills and band shirts.<br />
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<b>Background</b><br />
Over the past couple of years, the math team at <a href="http://www.zbths.org/ntzb" target="_blank">my school</a><b> </b>worked together to put a four-year curriculum in place that's closely aligned to the <a href="http://www.act.org/standard/planact/math/" target="_blank">ACT college readiness standards in mathematics</a>. The skill that my students are currently working on is XEI 602:<br />
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<i>"Write expressions, equations, and inequalities for common algebra settings."</i><br />
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We wrote a ton of problems related to each skill. For this particular skill, we wrote problems such as the following:<br />
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<i> </i><span style="font-size: xx-small;">(And actually, that should be 46 cakes, not 44. Typo. Oops.)</span></div>
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<i> </i><br />
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To supplement this skill, I thought of a different way to present this type of problem. Instead of spelling out the necessary mathematical information in a word problem, I wanted to present a more realistic situation and have the students work a little bit more to dig up the mathematics of what was happening.<br />
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So I thought of the following scenario: Suppose you were selling a few different items and keeping track of your sales on a sheet, such as this:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp2J_jfTM4KNGSZrao4hBMlPNvemR5N4QZ7hyphenhyphenMIJYy_ImjRbmXxdz9jOCa02L7IRSZOBu8TX-8Pa-COsi_XdNuRdT-ncyMkbSrdZTw6GtEUlwjz0NSwPV-Hke1WEqOHiAQYPWoTDyFo4ev/s1600/Screen+Shot+2013-09-09+at+12.59.14+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="467" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp2J_jfTM4KNGSZrao4hBMlPNvemR5N4QZ7hyphenhyphenMIJYy_ImjRbmXxdz9jOCa02L7IRSZOBu8TX-8Pa-COsi_XdNuRdT-ncyMkbSrdZTw6GtEUlwjz0NSwPV-Hke1WEqOHiAQYPWoTDyFo4ev/s640/Screen+Shot+2013-09-09+at+12.59.14+PM.png" width="640" /></a></div>
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And then, suppose you accidentally spilled coffee all over it:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXzgpC0HcdiA2E7TkiuGpwGwJz8AYhTOozxnB0s87m8IB7vym04WQJ_bTDIDWB0rCmA99iWZw0xfqeUcYQsNU95SOsiztH5HAzEW5nlY79pzkTHBKJSIoNTL__axMRsFWbmVHkkr7yW5yh/s1600/Screen+Shot+2013-09-09+at+1.00.48+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="452" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXzgpC0HcdiA2E7TkiuGpwGwJz8AYhTOozxnB0s87m8IB7vym04WQJ_bTDIDWB0rCmA99iWZw0xfqeUcYQsNU95SOsiztH5HAzEW5nlY79pzkTHBKJSIoNTL__axMRsFWbmVHkkr7yW5yh/s640/Screen+Shot+2013-09-09+at+1.00.48+PM.png" width="640" /></a></div>
<br />
Some of the information is lost! How could we figure out the information that was ruined by the coffee spill? (Obviously, there isn't enough mathematical information in the above example, which is purely for show. But given the right info, this becomes a challenging math task. Also, as it turns out, it's pretty challenging to simulate a coffee spill. And ink is pretty resilient these days.)<br />
<br />
<br />
<b>The Task</b><br />
I spent some time thinking about what product(s) to include on the sales sheet that I was going to spill coffee on. One night, I was folding laundry and I came across my official "The Returners" t-shirt. My brain was like, "BAM. T-shirts!"<br />
<br />
I messaged Lauren to pretty much say, "Hey, I'm using your band in a math problem!" And she basically replied, "Cool! Can I do anything to help?" And <i>actually</i> involving her hadn't really occurred to me, but she was totally willing to assist and I couldn't turn that chance down.<br />
<br />
I put together a "sales sheet" listing three different types of Returners shirts, with information about prices and inventory. Then I spilled coffee on it:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbt54FX9LaaslFeUAF8aI-bIH1Cs22koGrsJ46fl7U4RgIH9yZzTmda0AYdSrKcmCmmEUfgJWIJKaHL-HUAz5BDhscz9XYKvCi0UVC6OJtUc3lXnhYqQZxDgZRpB5Gg4Sf7qGu2NmaArng/s1600/Photo+Sep+16,+7+11+52+AM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbt54FX9LaaslFeUAF8aI-bIH1Cs22koGrsJ46fl7U4RgIH9yZzTmda0AYdSrKcmCmmEUfgJWIJKaHL-HUAz5BDhscz9XYKvCi0UVC6OJtUc3lXnhYqQZxDgZRpB5Gg4Sf7qGu2NmaArng/s640/Photo+Sep+16,+7+11+52+AM.jpg" width="640" /></a></div>
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Lauren included the following e-mail, and also sent along a picture of a few of her band's shirts (rolled up neatly into little shirt burritos):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSQdQt_iks_TGHyPO3aEMgl97InscrQqbSRkqT80xzPba_qqwvRRVBQseUGO7ByX3WG2zmhXcLLfUczr6ZKUqieAY90iRNHvLoKs0KGfwiiczzxt1sXsBlXwil-5LdsKg9GY_fTfH7tKyr/s1600/Screen+Shot+2013-09-17+at+2.24.43+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSQdQt_iks_TGHyPO3aEMgl97InscrQqbSRkqT80xzPba_qqwvRRVBQseUGO7ByX3WG2zmhXcLLfUczr6ZKUqieAY90iRNHvLoKs0KGfwiiczzxt1sXsBlXwil-5LdsKg9GY_fTfH7tKyr/s640/Screen+Shot+2013-09-17+at+2.24.43+PM.png" width="640" /></a></div>
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<br />
Armed with the above information, my students were set to the task of helping Lauren figure out how many shirts were sold at her most recent show.<br />
<br />
<br />
<b>How The Task Went</b><br />
I put the students in groups of three for this task. They were each given a copy of the e-mail, the ruined sales sheet, and the photo of the remaining shirts. I also gave them a <a href="http://www.scribd.com/doc/169161717/Return-to-Sender-Worksheet" target="_blank">worksheet</a> with a few questions that each group member needed to contribute to.<br />
<br />
My first period jumped into this task right away. There were a lot of good conversations going on at the start: they were looking through the documents, figuring out what information was important, and talking about how they were going to represent each type of shirt as variables in equations.<br />
<br />
There was debate over how to represent "blue" versus "black," since both started with the same letter. Some students decided to use <i>b</i> for one and <i>bl</i> for another, but soon found that there was still no distinction (since both colors start with the letters <i>bl</i>). One student finally suggested using <i>k</i> for black, which certainly helped things.<br />
<br />
Students were able to figure out important bits and pieces of information needed to set up an equation to represent the total sales: from the sales sheet, they found the total money made and the cost of each shirt. From the picture, they were able to determine that there were 5 blue shirts and 3 gray shirts that went unsold from the original inventory.<br />
<br />
Where we ran into trouble was figuring out how to <i>actually</i> set up equations representing the total sales. Groups during my first period class initially set up their equation as:<br />
<br />
<div style="text-align: center;">
5.25<i>k</i> + 4.75<i>b</i> + 4.50<i>g</i> = 397.25</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
where <i>k</i> represents black shirts, <i>b</i> represents blue shirts, and <i>g</i> represents gray shirts. The above equation was close, but incorrect; students from my first period continued working through the problem using this equation and ended up getting answers that didn't make sense in the context of the problem (e.g. they got non-integer values when they solved for <i>k</i>, <i>b</i>, and <i>g</i>).</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
They were on the right track, but they didn't account for the fact that a few shirts went unsold; this was kind of the "tricky" part of the task and led to a lot of frustration among my first period students. I let them have some time to try and sort things out on their own and dropped a few hints to try and point them in the right direction. Eventually, I saw it was going to be best to stop and have a quick whole-class discussion about the unsold shirts.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
We talked about thinking of <i>k</i>, <i>b</i>, and <i>g</i> as the number of shirts that were <i>originally in the inventory</i> as opposed to the number of shirts <i>that were sold</i>. I asked the students to look through their documents again and tell me what they could find about the number of shirts that were sold and the number of shirts that were still left. We talked about what expressions we should write to represent the number of shirts <i>that were sold</i>. Eventually, we came up with the following:</div>
<ul>
<li>The black shirts were sold out, so <i>k</i> black shirts were sold.</li>
<li>There were 5 blue shirts remaining, so <i>b - </i>5<i> </i>blue shirts were sold.</li>
<li>There were 3 gray shirts remaining, so <i>g</i> - 3 gray shirts were sold.</li>
</ul>
The students adjusted their equations to reflect an accurate model of the total sales:<br />
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After my first period class, I adjusted my lesson plan so that we talked about the correct expressions for shirts sold toward the beginning of the activity. This adjustment made things go more smoothly in my other three classes. Although, in my second period class, students were having trouble with question #2, which required them to re-write their equation from question #1 in terms of one variable. We stopped to have another conversation about what to do.<br />
<br />
I asked students to re-read their e-mails and to look specifically for relationships between the different types of shirts and how many of each kind there were. The students noticed the following info:<br />
<ul>
<li>There were twice as many black shirts as either the blue or gray shirts.</li>
<li>The number of blue shirts was the same as the number of gray shirts.</li>
</ul>
From here, we talked about how to use this information to write a few small equations that would help us with question #2. The students came up with the following equations:<br />
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<br />
Once we were armed with these equations, we were able to go back to our total sales equation from question #1 and use substitution to re-write it in terms of one variable:<br />
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<br />
<br />
I made another adjustment to my lesson plan for my third and fourth period classes to include a discussion about representing these relationships toward the beginning of the task as well. With this guidance, students were able to successfully determine how many shirts had been sold:<br />
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<b>What I Would Change Next Time</b><br />
This task was given to the students while we were in the middle of our mini-unit on writing equations and expressions based on information from word problems. After doing this task and reflecting on how things went, I think this task has a lot of merit as one of two things:<br />
<ol>
<li>A guided task at the beginning of, or during, the unit, with appropriate scaffolding included; or</li>
<li>A performance task at the end of the unit.</li>
</ol>
My students were resilient and we were able to have a lot of good mathematical conversations during this task. I could also tell there was frustration stemming from confusion about what to do at each problem. I think this might have been the result of not yet having enough practice with the skill, and I also think I didn't provide an appropriate amount of scaffolding, originally. In my later classes, I made adjustments and discussed some important aspects of the problem at the beginning of the task; this seemed to help students successfully solve the problem.<br />
<br />
In the future, I would probably embed further scaffolding and questions into this activity. For instance, I would probably ask the students:<br />
<ul>
<li>What is the problem you are being asked to solve? What information are you supposed to determine?</li>
<li>Choose a variable to represent each different type of shirt. Write expressions to represent the amount of each shirt that was sold.</li>
<li>From the e-mail, what information can you determine about the number of shirts that Lauren originally had? Write equations that represent these relationships.</li>
</ul>
By having the students doing this work first, the rest of the task would probably go more smoothly as is.<br />
<br />
I might also change the prices and the types of merchandise on the sales sheet in the future; a few students looked at the prices and thought, "why didn't they just sell shirts for $5? It'd be easier to make change." In reality, the shirt prices are actually more like that. I made each shirt a different price; otherwise, the task would have been pretty easy to solve. Next time, I'd probably include other merchandise such as CDs, buttons, etc. and let the prices reflect something that would <i>actually</i> be charged at a show.<br />
<br />
Overall, I was pleased with this task. I certainly learned a lot, and I hope to come up with even better tasks in the future and I continue trying to figure out how to do PrBL!<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com3tag:blogger.com,1999:blog-3248548197809593245.post-27098153961096864612013-09-06T14:23:00.001-05:002013-09-06T14:25:54.902-05:00Mario: Life or Death? (A Math Task)I love math. I also love video games. So when I find a way to combine the two, I'm like, super-mega-happy.<br />
<br />
We're in the earlier stages of a unit on graphing, and I want my
students to get used to the idea that they'll need to interpret &
analyze figures, diagrams, etc. and make predictions or generalizations
based on solid mathematical reasoning. This is a skill that I believe is
crucial for a student's success in Pre-Calculus and Calculus (and
beyond). <br />
<br />
This week, my Advanced Pre-Calc students did a one-period math task I developed called, "Mario: Life or Death?" It wasn't anything terribly complex or flashy, but we did get at a few important ideas about graphing, symmetry, and making predictions from what we know <i>conceptually</i> rather than <i>numerically</i>.<br />
<br />
(Also, after I came up with the idea for a Mario-based problem, I found out about this <a href="http://simplifyingradicals2.blogspot.co.uk/2013/02/super-mario-bros-results.html" target="_blank">Mario problem from Nora Oswald</a> that's way, <i>way</i> better than what I did. I was actually inspired to follow a similar format after finding it.) <br />
<br />
I was taking (or trying to take) a <a href="http://blog.mrmeyer.com/?p=16470" target="_blank">3-Act Math</a> approach with this math task. It definitely still needs some tweaking all around, but considering I'd never tried anything of the kind before, I was pretty pleased. (I don't know how <i>truly</i> 3-Act-ish this task actually is, but, eh.)<br />
<br />
<br />
<b>The Opening</b><br />
I played this video for the students:<br />
<br />
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<br />
The students also had the first two pages of <a href="http://www.scribd.com/doc/166091641/Mario-Life-or-Death-Worksheet" target="_blank">this worksheet</a> (I purposefully withheld pages 3 and 4 until later).<br />
<br />
As students watched the video, I had them think about what questions came to mind. They came up with some pretty good ones:<br />
<ul>
<li><i>Will Mario make it to the other side?</i></li>
<li><i>When should Mario jump to make it to the other platform?</i></li>
<li><i>What is Mario doing wrong that's not letting him jump across the cliff?</i></li>
<li><i>When does Mario reach the apex of his jump?</i></li>
<li><i> How can we calculate what height he needs to jump to make the distance?</i></li>
</ul>
And some pretty good non-math questions, too:<br />
<ul>
<li><i>Why doesn't he use the chain to make it across?</i></li>
<li><i>Did the player hold down the A button long enough?</i></li>
</ul>
So right away, the students came up with some important concepts, including distance, height, and particularly "reaching the apex" of the jump. These were important things to notice, and were very closely related to the problem I posed (which a few of them nailed):<br />
<i> </i><br />
<div style="text-align: center;">
<i><b>"Will Mario make it across on the 3rd jump?"</b></i></div>
<br />
<br />
<b>Working On the Problem</b><i></i><br />
Next, the students were asked to come up with a few ideas. What information did they think would be needed to solve the problem? I gave them a little time to think of a few ideas on their own, and then they shared with each other. A couple of examples from their work:<br />
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<br />
With some group discussion, we pulled together some important concepts to think about: the distance between platforms, the height of Mario's jump, and in a few cases, the ideas of "arcs," "parabolas," and "symmetry."<br />
<br />
I had the students explore these ideas further by having them create sketches of what they thought the path of Mario's jump looked like:<br />
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We talked about the shape of Mario's jump. The words "semicircle" and "arc" came up a lot. One or two students already knew the word "parabola."<br />
<br />
I asked the students to think about what they knew about arcs; what is their structure? How do they behave? Particularly, I asked them if they've ever seen any "lopsided" arcs. The students said no; arcs are made up of "mirror images," or are "symmetrical." I asked them what they meant by that. They replied that an "arc" can be cut into two halves that are mirror images of each other; particularly, the arc is cut at the "highest point" into two congruent halves.<br />
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The students were asked to draw the path of Mario's jump again, this time on a grid. They were also asked to take the observations from their discussion into account:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcK8-lnshRon05G6FZ6LoHspNEh-nwFGWorNfmTzedcZ-s-fV8bXylTvu8DrVqiz-g2Qd8XsyBP895FilrtkprfiPBzjbxcQ2qZucOum_2shlaUXxo0Pl5aFlFAiB-h50eYZMR5kY6Scsk/s1600/Photo+Sep+05,+12+25+55+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcK8-lnshRon05G6FZ6LoHspNEh-nwFGWorNfmTzedcZ-s-fV8bXylTvu8DrVqiz-g2Qd8XsyBP895FilrtkprfiPBzjbxcQ2qZucOum_2shlaUXxo0Pl5aFlFAiB-h50eYZMR5kY6Scsk/s320/Photo+Sep+05,+12+25+55+PM.jpg" width="320" /></a></div>
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After looking at the gridded drawings, the students had a few moments to think about how their observations about Mario's jump could be useful in predicting the outcome. They wrote their thoughts down:<br />
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After giving them time to think about how to use their observations to predict whether or not Mario would make the jump, I gave them pages 3 and 4 of the worksheet. Using the graphs of Mario's jumps, I asked the students to make their predictions and to give their mathematical reasoning:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNhMgnAMzgu5t-E4b3xQMTZWxRg0MUpEzrM8t1pIv5TbYC2I24hsyUt4SVm4AV0Vt9dwVSlX1_me9rfMYvlgdqQNyWLv-qcAsTbP72J9QDZr0mXeznl-6ihUGWlw82bAe_KpvvBMnYFQr1/s1600/Photo+Sep+05,+12+29+35+PM.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNhMgnAMzgu5t-E4b3xQMTZWxRg0MUpEzrM8t1pIv5TbYC2I24hsyUt4SVm4AV0Vt9dwVSlX1_me9rfMYvlgdqQNyWLv-qcAsTbP72J9QDZr0mXeznl-6ihUGWlw82bAe_KpvvBMnYFQr1/s400/Photo+Sep+05,+12+29+35+PM.jpg" width="400" /></a></div>
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Everyone shared their predictions and gave their reasoning. The consensus was that Mario <i>would</i> make his jump, but just barely.<br />
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<br />
<b>The Thrilling Conclusion</b><br />
After everyone made their predictions, the moment of truth arrived. I played the solution video:<br />
<b> </b><br />
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dxSEkU2mmO3_u6SsubJqEyz00hxzv3grnChfJmbxN5CWHKILadFoCcVSk0BFcY2bdVrrTPqXuyTnonkXsRl6w' class='b-hbp-video b-uploaded' frameborder='0'></iframe></div>
<b> </b><br />
So the students had predicted correctly! They seemed a bit amazed at how accurate their predictions were -- not only had they figured Mario would make his jump, but many of them said specifically that Mario would <i>barely</i> make his jump, which is what happened. They had made a mathematical prediction using conceptual observations about the path of Mario's jump, and very minimal use of numbers or calculations.<br />
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To wrap things up, we tied our discussion back to the idea of symmetry. The type of symmetry used in this problem was just one <i>type</i> of symmetry (even symmetry, and arguably symmetry about the y-axis). We talked about other types of symmetry besides the type used in this problem. The students took down some notes for their problem sets they were working on from our textbook.<br />
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I hope that the students came away from the lesson with a bit of a deeper understanding of symmetry and how to use it to make predictions. In the larger picture, I hope students started to see the usefulness of applying what they know and observe in order to gleam new information. As I've said before, there's <a href="http://brennemath.blogspot.com/2013/08/mathspotting-because-math-hides-in.html" target="_blank">so much more to math</a> than just calculations and number-crunching.<br />
<br />
This was totally the highlight of my week. However, I know that this task is far from perfect. It could certainly go deeper and touch on quadratics, graphing equations, and so forth. I'm almost certain this task could be made into something richer than its current form. I'll definitely be taking another look at it after some time passes and I'm free enough to sit down and make revisions.<br />
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But in the meantime, I welcome any and all feedback/comments/insults from any readers out there. Don't be shy!Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com11tag:blogger.com,1999:blog-3248548197809593245.post-16523998943817761422013-09-02T11:27:00.001-05:002013-09-02T13:27:48.122-05:00Week 1: Why My First Day Activity Didn't Go At All As I Had Hoped (and Why That's Awesome)Phew, the first week has come and gone and I found myself utterly exhausted on Friday. Thank goodness for the holiday weekend; I've been able to get more sleep in the past 2-3 days than I have in quite a while.<br />
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As part of the kickoff to our school year, I had my seniors work on an "opening day" activity that I lovingly borrowed/blatantly stole from <a href="http://twitter.com/MrLeNadj" target="_blank">Nadji</a> (who blogs at <a href="http://physixcoolisms.blogspot.com/" target="_blank">Physix Coolisms!</a>) that involves grids, writing your name (a <i>lot</i>), and using that data to generalize a pattern.<br />
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The activity I snagged is called "What Is Math?" and is described by Nadji from 4:25 to 12:30 of <a href="https://www.bigmarker.com/GlobalMathDept/aug06" target="_blank">this First Day of School Activities presentation from Global Math Dept</a>. I won't re-post the entire activity here, but basically the aim of the activity is to challenge students' perceptions of what it means to do math.<br />
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<b>The Activity</b> <br />
The first part of the activity has students answer the following questions:<br />
<ol>
<li><i>What is math? What does it mean to you?</i></li>
<li><i>List 7 mathematical words or phrases that come to mind when doing math. </i></li>
</ol>
Often, students respond to these questions with the mindset that "doing math" means working with numbers and calculations and equations. <br />
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After answering these questions, students then <a href="http://s3.bigmarker.com/f2c7e467189d1bbf46a1ece955c8e900e356aeae-1375835991822/presentation/nadji-opening-day-GMD/slide-4.png" target="_blank">fill out several square grids by writing the letters of their name over and over again</a>.<br />
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After doing that, the students shade the first letter of their first name and then <a href="http://s3.bigmarker.com/f2c7e467189d1bbf46a1ece955c8e900e356aeae-1375835991822/presentation/nadji-opening-day-GMD/slide-5.png" target="_blank">fill out a table</a> to record the patterns that show up.<br />
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From there, students are asked to make predictions about the patterns, such as:<br />
<ul>
<li>Predict what pattern would appear in a 41x41 grid.</li>
<li>Predict how the patterns would be affected if the second letter of each name was shaded instead.</li>
<li>Predict how the patterns would be affected if students started by writing their name in the bottom right corner and filling out the grid backwards.</li>
</ul>
And so on.<br />
<br />
At the end of the activity, students are asked the beginning two questions again; by this point, the hope is that students will start to see that math is much more than just working with numbers and calculations and equations. There is much more to mathematics: finding patterns, making generalizations, predicting unknown events, thinking critically, etc.<br />
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<br />
<b>How Things Went:</b><br />
Before I get into this, one side (yes, <i>side</i>, not <i>snide</i>) comment: I had my students fill out grids up to 10x10. If I do this activity again, I might have them go up to 12x12. I have many students with names that are 6, 7, or 8 letters long, and their patterns don't really start to become apparent until the grids get bigger. As I checked in on students and looked through the tables they were filling out, it seemed to me that the "pattern of the patterns" would be more apparent if they had more data. Something to think about for next time. <br />
<br />
At any rate, student responses to the opening two questions went pretty much as I expected. Many students came up with responses like "math is the study of numbers," or "math is the tool of Satan," and so forth. The lists of 7 mathematical terms often included "addition, subtraction, multiplication, division, square root, equation, numbers," and the like.<br />
<br />
I decided to collect answers to the first two questions via <a href="http://www.socrative.com/" target="_blank">Socrative,</a> so I could quickly generate a bunch of text and then dump them into a <a href="http://www.wordle.net/" target="_blank">Wordle.</a>
I thought it would be cool to generate a visual snapshot of student
responses from before and after the activity so I could compare.<br />
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Here is the "before" Wordle:<br />
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As you can see, there's a great deal of "number-ish, calculation-y" stuff. I expected to see this.</div>
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Based on what I was seeing from the students as they were working on the activity and the conversations they were having (with each other and with me), I expected to see a dramatically different Wordle from the post-activity responses. After all, they were noticing patterns, making predictions about how patterns would look in grids that were far larger than they had time to fill out, and working together to describe a "rule" for making such a prediction. They weren't <i>really</i> doing "stuff with numbers."</div>
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So, here's how the post-activity Wordle turned out:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEief8F3era67KNDSO-hKllVt3c26RXgr4ioISi8Hr0gRX-LzC8GrN4H3bvoZtxyN1K3uFXJO_oDTxpytj8mtwXrx7Qu1ZXQAVF5OCNs0AhWgRxUCEccusCIlNj_cz_sf9hp4vqCphk8-gXw/s1600/Screen+Shot+2013-09-02+at+11.07.34+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEief8F3era67KNDSO-hKllVt3c26RXgr4ioISi8Hr0gRX-LzC8GrN4H3bvoZtxyN1K3uFXJO_oDTxpytj8mtwXrx7Qu1ZXQAVF5OCNs0AhWgRxUCEccusCIlNj_cz_sf9hp4vqCphk8-gXw/s640/Screen+Shot+2013-09-02+at+11.07.34+AM.png" width="640" /></a></div>
So uh... um... not really all that different. I mean, "patterns" showed up a lot more in this one, but there was still a bunch of "number-ish, calculation-y" stuff.<br />
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I'll admit, at first I was a little bummed that I seemingly hadn't changed very many minds or shifted very many paradigms after doing this activity.</div>
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But then I thought about it. And I became okay with it.</div>
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<b>In fact, it's actually pretty awesome that I didn't change their minds so easily, and here's why:</b></div>
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This becomes a new challenge for me. This allows me to set a goal for myself. I want my students, by the end of the year, to understand that there's a lot more to mathematics than just crunching numbers and solving numerical problems.</div>
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Math is recognizing patterns and trends. Math is making use of those recognitions to make predictions. Math is critical thinking.</div>
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Math is art. Math is visual, spatial, tangible.</div>
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Math is <i>freaking everywhere </i>and <i>freaking awesome</i>. </div>
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I don't get to spend just one day trying to convince my students of this. I get to spend <i>an entire year</i> trying to convince my students how super-cool math is. I have a lot of convincing to do, but that's okay with me. I want to earn it.</div>
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That's one thing I learned from doing this activity. That's one thing this activity has given to me: a theme for this year: <a href="http://brennemath.blogspot.com/2013/08/mathspotting-because-math-hides-in.html" target="_blank">Math is </a><i><a href="http://brennemath.blogspot.com/2013/08/mathspotting-because-math-hides-in.html" target="_blank">freaking everywhere</a> </i>and <i>freaking awesome</i>.</div>
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It's going to be a great year. </div>
Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com3tag:blogger.com,1999:blog-3248548197809593245.post-9557539156296075972013-08-28T12:51:00.000-05:002013-08-28T20:36:34.057-05:00Reflections From #precalcchat: Pre-Calculus SequencingI love <a href="http://www.twitter.com/" target="_blank">Twitter</a> chats with other teachers. It's a great way to make connections. It's a great way to get insight, ideas, and resources. It's also a fantastic opportunity to reflect on your own practice and to improve what you're doing in the classroom.<br />
<br />
The <a href="https://www.bigmarker.com/communities/GlobalMathDept/about" target="_blank">Global Math Department</a> hosts <a href="http://mathchats.pbworks.com/w/page/68161831/Twitter%20Math%20Chats" target="_blank">several weekly Twitter chats</a> for math teachers on a variety of topics. Since I teach Pre-Calculus, I dropped in on the first <a href="http://storify.com/untilnextstop/precalcchat?utm_medium=sfy.co-twitter&utm_source=t.co&awesm=sfy.co_rD0K&utm_content=storify-pingback&utm_campaign=" target="_blank">#precalcchat</a> of the school year last week; thanks to <a href="http://www.twitter.com/untilnextstop" target="_blank">Mimi</a> (<a href="http://untilnextstop.blogspot.com/" target="_blank">I Hope This Old Train Breaks Down...</a>) and <a href="http://www.twitter.com/MrLeNadj" target="_blank">Taoufik Nadji</a> for hosting. Couldn't spend much time, but the topic of conversation captured my interest:<br />
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I loved that thought. It made me stop and think about how I sequence my Pre-Calculus course and why.<br />
<br />
I start with Graphing and Functions first. To me, it's important for students to understand the basics of interpreting graphs of functions and becoming fluent with moving between different representations of a function (graphs, tables, equations). I find this to be a particularly vital theme that I want to drive home with my students, especially those who will be going on to AP Calculus or Calculus I/II in college.<br />
<br />
Next, I follow a pretty standard sequence of Quadratics/Polynomials, Rational Functions, Exponential Functions, Logarithmic Functions, Trigonometric Functions, and Analytical Trigonometry. Again, I focus on these topics in particular to prepare my students for success in an AP Calculus course. Other topics such as Analytical Geometry, Series & Sequences, Polar Systems of Coordinates, Conics, etc. come afterward as time allows.<br />
<br />
The other chatters all had brilliant things to say, so naturally I felt like I'm probably doing everything wrong (or maybe just some things wrong, and other things not-as-wrong).<br />
<br />
When discussing how Pre-Calculus can seem like a re-teaching of Algebra II to students, <a href="https://twitter.com/crstn85" target="_blank">Tina C</a> (<a href="http://drawingonmath.blogspot.com/" target="_blank">Drawing On Math</a>) mentioned that her school <i>starts</i> with Trigonometry for that exact reason.<br />
<br />
This was an interesting idea to quite a few of us: do Trig first semester, slowly build up conceptual understanding of the unit circle, graphing, transformations, identities, etc. Then, move into the other different functions second semester.<br />
<br />
The more I think about doing Trig first, the more appealing it seems to me. I've always found that I never seem to have enough time to really properly teach Trig and I need to either rush a few things or cut some other stuff out. I think I probably always had the notion that Trig is "more difficult," and somehow it made sense to put the "harder stuff" at the end of the year. (That's <i>excellent</i> reasoning, isn't it?)<br />
<br />
But really though, Trig is a bit of a stand-alone topic. It could go anywhere in the course sequence. There are certainly some underlying concepts that can be applied to other functions: graphing, transformations, moving fluently between representations, and so on. I usually think of these concepts as having to be taught and mastered <i>before</i> doing Trig, as if Trig is the "CHALLENGE MODE" of working with functions in Pre-Calculus.<br />
<br />
Who's to say we can't use Trig to <i>teach</i> these concepts instead? Maybe my students would have greater success with Trig if I did it at the beginning of the year, built the concepts slowly with appropriate scaffolding, while still equipping students to be successful in working with other functions. I may have to try it out one of these years. (I already have this year mapped out -- maybe next year?)<br />
<br />
Anyway, some great food for thought. <br />
<br />
I'm looking forward to more of these chats this school year, and hopefully I'll find time to continue blogging & reflecting on what I take away from them. Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com2tag:blogger.com,1999:blog-3248548197809593245.post-48299118002150635782013-08-21T10:05:00.000-05:002013-08-21T10:05:00.469-05:00Week Zero: Realizing I Might Actually Know StuffIt's Week Zero. School Year Eve. The last few days of summer before I get to go back into the classroom and spend the next nine months convincing teenagers that math is <i>freaking awesome</i>.<br />
<br />
I'm a teacher mentor this year, which still seems crazy to me because I'm only four years into this profession myself. On Monday, I went to an all-day mentor training session to learn about my role and responsibility as a mentor. A lot of the information was about what I had expected: the mentor wears many different hats, has to build a relationship of trust with the mentee, can learn just as much about teaching from the mentee as the mentee does from them, etc. and so on. We talked about how to have positive conversations with our mentees, how to listen and to provide feedback, and best mentoring practices in general.<br />
<br />
We also got toys and candy, which was super cool:<br />
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<br />
One thing that struck me from the mentor training was what distinguishes a good mentor from a not-so-good mentor: the desire to <a href="http://brennemath.blogspot.com/2013/08/never-be-fully-satisfied.html" target="_blank">keep getting better as a teacher</a>. Good mentors know that they still have things to learn about teaching, and no matter what the difference in experience is, they can learn a lot from their mentees. (I'm pretty convinced that I'm going to learn more from my mentee than my mentee is going to learn from me.)<br />
<br />
I was reminded of this the next day (Tuesday) when I attended the first-day morning session of new teacher orientation. I sat with my mentee throughout the morning as we introduced ourselves and learned various things about the teacher-mentor program. We had time to talk about the upcoming school year and I was able to answer some questions about curriculum and how we do things at our school.<br />
<br />
The experience made me think back to my first Week Zero in our district, when I went through new teacher orientation. I remember feeling excited and nervous about my first year of teaching. I also remember thinking that I was probably going to make a lot of mistakes, I was going to have to learn from them, and there was <i>so so much</i> about teaching that I didn't know yet.<br />
<br />
I had the same excited, nervous feeling this week. I still feel like there is <i>so so much</i> about teaching that I don't know. But, in the act of answering my mentee's questions, I was struck by another thought: I actually, maybe, perhaps, <i>do know</i> <i>stuff about teaching now</i>. I had never really thought about it until someone else was asking me. When I was answering my mentee's questions, I really had a lot to say. I had a place of experience to speak from. <i>Holy crap, I have experience.</i> <i>And it might even be useful to someone else.</i><br />
<br />
That might be my important realization from this week: There are many things about teaching I still don't fully know. But I'm also starting to understand how much I <i>do</i> know about teaching. Maybe I'll actually be a decent mentor.<br />
<br />
Anyway, back to work! Students come back next week!Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com0tag:blogger.com,1999:blog-3248548197809593245.post-32590429969270753582013-08-18T17:00:00.003-05:002013-08-18T17:03:33.048-05:00Never Be (Fully) SatisfiedThe past few days, I've been reflecting on how much time I spent this summer working on writing and tweaking curriculum for the new school year. It's not exactly a new activity for me -- I pretty much write and tweak curriculum <i>every</i> summer -- but I think I probably got more done this summer than I've ever managed to.<br />
<br />
I actually fleshed out two different curriculum maps with topics & aligned standards (first attempt at aligning Common Core, so probably lots of mistakes). I'd never made curriculum maps with a great level of detail before, and I'm pretty sure I'm going to be very thankful I did so this summer.<br />
<br />
I also spent a <i>lot</i> of time this summer working on incorporating more Problem-Based Learning (PrBL) tasks & lessons into the curriculum (one such idea I had is detailed <a href="http://brennemath.blogspot.com/2013/07/battleship-graphing-equations-of-circles.html" target="_blank">here</a>; feedback is more than welcome!). I teach at a <a href="http://www.newtechnetwork.org/" target="_blank">New Tech Network</a> school, so a rigorous PrBL curriculum is my goal. I've spent hours and hours looking for ideas, researching, thinking, scribbling in my notebook (particularly for those middle-of-the-night ideas), typing pages of details, and probably making my wife very annoyed that I was spending so much time working. I hope the result is that my students do some really awesome, really meaningful learning this year. <br />
<br />
Another goal of mine is to learn more about Common Core (I admittedly am still a novice), so when my principal e-mailed the staff earlier this summer about attending a Common Core workshop in September, I was all like "MEMEMEMEMEME!!" So, I'm excited to go, learn some more about Common Core, and hopefully take away valuable knowledge that I can incorporate into my professional practice.<br />
<br />
And the idea of improving my professional practice is something I've been thinking about over the past few days.<br />
<br />
I've found myself thinking a lot about all the things teachers do to try and improve their teaching. I see many teachers who I follow on Twitter talk about all the conferences they attend and share what they've learned. I have several friends who are enrolled in masters programs, learning more about educational technology, developing curriculum, or otherwise broadening their skill sets as educators. I've thought about the things I've done each summer since I started teaching: working on curriculum, participating in the professional community, working on my own masters, and constantly thinking (and often worrying) about how I can be a better teacher.<br />
<br />
And as I thought about all of this, I realized something: I'm not sure I ever want to be <i>satisfied</i> with the kind of teacher that I am.<br />
<br />
I'm sure not satisfied with my teaching right now. Frankly, I'm not that great at it. (Sure, I'm funny, handsome, irresistibly charming, and <i>very</i> humble; but from a pedagogical standpoint, those traits can only carry me so far.)<br />
<br />
But I don't think I want to <i>ever</i> be fully satisfied with my teaching, not even after I've been teaching for thirty (forty? fifty?) years. Sure, I want to feel <i>happy </i>about my teaching, which I think is a different thing. But not <i>satisfied</i>.<br />
<br />
I think it's probably easier to feel this way now, since I'm only going into my fifth year. I <i>know</i> that I have a lot more to learn about teaching. Any fool can see that. There are roughly eleventy billion areas where I can to improve my teaching. I have rather lofty goals for myself this year. I might not meet them all this year, but that just means I'll regroup next summer and try again the following year. And the following year. And the following year. And so on.<br />
<br />
But when I've been teaching for a few decades, I don't know how easily I'll still see all of that. I don't know if I'll still be this enthusiastic about improving my craft or if I'll be like, "meh, I've been teaching for thirty (forty? fifty?) years, I'm awesome enough." I don't like that idea. I really hope instead that I'll always want to be a better teacher than I was the year before. Even if it's just a <i>teensy</i> bit better. My students deserve that much, I think.<br />
<br />
I talked about this with my wife the other evening. She understood where I was coming from, and noted that this is true about many professions. I mentioned that I was (and am) nervous about meeting my new students on the first day. She said one of her past supervisors once told her that's normal; "that means you care." And my feeling nervous doesn't really stem from being scared about meeting a new group of people, but more from <i>really really wanting to be a better teacher this year than I was last year</i>. I don't want to let these kids down.<br />
<br />
Summer is great. It's a time when teachers can work on improving themselves and do what they can to make the next school year better than the last one. I spent a <i>lot</i> of time this summer working on that. I always want to be doing that. I want to be happy with who I am as a teacher. But I also think that I want to never be satisfied. Maybe <i>mostly</i> satisfied. But not <i>fully</i> satisfied.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com4tag:blogger.com,1999:blog-3248548197809593245.post-44505422408601038352013-08-09T16:02:00.001-05:002013-08-09T16:37:43.245-05:00For the Interns and the First-Years: 2013 EditionIt's almost time for another new school year to begin! Some of you have already started. Some of you start Monday. Some of you still have a few weeks left and are itching to go.<br />
<br />
Last year, I wrote a post with some <a href="http://brennemath.blogspot.com/2012/08/for-interns-and-first-years.html" target="_blank">advice for new teachers and students entering their internship</a>, and received many positive responses. I figured it might not be a bad idea to re-post it this year, with a few alterations based on comments I've received and my own newly-learned experiences.<br />
<br />
Other teachers, please feel free to chime in with your own advice for first-years and interns in the comment box below! I would <i>especially</i> love to hear from second-year teachers with what advice they would give to a new teacher.<br />
<br />
Without further ado, "For the Interns and the First-Years: 2013 Edition." Eleven pieces of advice that I have for interns and new teachers (that I never followed myself because I was more of an idiot then than I am now):<br />
<br />
<br />
<b>1. This year is not going to kill you.</b><br />
<br />
This is the thing you need to know, first and foremost: you are going to survive this year. I mean, it's not <i>really</i> a life-or-death situation, but it sure as hell will feel like it sometimes.<br />
<br />
You're going to work very hard, you're going to have some awesome
moments along the way, and some not-so-awesome moments as well.<br />
<br />
This is a year to discover who you are as a professional and as an adult.<br />
<br />
You're not going to do everything right. New teachers never do. Veteran teachers never do. What matters is that we try to improve, all the time.<br />
<br />
We try new things. We refine old ideas. We reflect. We seek feedback from others. It's a process that continues long after you've left college. Some of us have been teaching for years and we're <i>still</i> just getting some things about this craft of ours figured out.<br />
<br />
Your most important skill as an educator isn't your ability to teach; your most important skill as an educator is your ability to learn.<br />
<br />
Learn what you can from this year. From yourself, from your colleagues, from your students. It's wonderful. It's possibly one of the most challenging and exhausting years you'll ever have, but it's worth it. All you can really do this year is your best, and grow from there.<b> </b><br />
<br />
<br />
<b>2. Use your support systems.</b><br />
<br />
There are going to be times when you feel very alone and overwhelmed. These are the times that you need to reach out to people.<br />
<br />
I called my mom so many times during my internship.<br />
<br />
I called, texted, and IMed (this was in the AIM days, mind you) other math ed students in my cohort. (They called, texted, and IMed me, too.)<br />
<br />
I vented to my professor (who I remain friends with to this day). I vented to other teachers.<br />
<br />
I vented to my girlfriend (who somehow thought it was still a good idea to marry me later on).<br />
<br />
If I hadn't had all of these people around to listen to me when I needed
to talk to someone (or ask a panicked question about what the hell a
"unit" was and how to plan it), I'd probably be locked up in a padded
room right now.<br />
<br />
Let your friends and family know that you might need to rely heavily on
their emotional support this year. You're going to need people to listen
to you and to advise you. Know that you have these people.<br />
<br />
Even reaching out to total strangers on the Internet (such as the <a href="http://mathtwitterblogosphere.weebly.com/" target="_blank">mathtwitterblogosphere</a> for instance) is an excellent way to lean on other teachers for support. Seriously, get a Twitter account. Find fellow teachers who are active in various online professional learning communities. Participate in weekly Twitter chats. Start a blog. You'd be amazed by the support and positivity you get from teachers you've never even met. <br />
<br />
Whatever you do, however you do it, whomever is part of it, just use the hell out of your support network.<br />
<br />
<br />
<b>3. Leave school at school.</b><br />
<br />
There are many times that teachers do have to take work home. Whether it's grading or spending extra time planning, we often end up working more than we probably should. That's just part of the job. But we do it.<br />
<br />
There are other times, though, when you can leave work, go home, and not
have to worry about tomorrow until tomorrow. This is when you need to
learn to <i>leave school at school</i>.<br />
<br />
During my internship, I would stay at school until 4:00, sometimes 5:00,
before packing up and going home. I spent this time grading, planning
new lessons, creating assignments, and so on. I'd be reasonably prepared
for the next day and decide it was time to head back to my apartment.<br />
<br />
Upon getting home, my brain wouldn't shut off. I kept thinking about <i>everything</i>:
wondering if I wasn't doing enough or if I was doing too much with my
lesson plans; wondering if I was going to do a great job the next day,
or totally bomb; wondering what other ways I could explain material to
the students, or what other activities I could possibly have them do;
and so on.<br />
<br />
It drove me crazy and kept me awake many nights.<br />
<br />
One way I found to cope with my inability to stop thinking about school
24/7 was to keep a notepad with me and write down ideas as they came. I
also wrote incredibly detailed lesson plans, sometimes pacing things
down to the minute. This let my brain have a chance to get everything
out and wind down. Gradually, I was able to get to a point where I could
leave school and flip off the "teacher switch."<br />
<br />
You can't always leave school at school; but when you can, learn how to do it effectively. It might take time (and experience), but it will come.<b> </b><br />
<br />
<br />
<b>4. Be realistic when you do take work home.</b><br />
<br />
A couple of teachers gave me a great rule of thumb when it comes to taking work home: "only take what you can carry in your hands." Realistically -- unless you're feeling unusually energetic or having some kind of manic episode (kidding, kidding) -- you're not going to get terribly much done when you get home.<br />
<br />
You're <i>probably</i> either going to be too tired to get much done, too busy with other after-school obligations, or just plain unmotivated. These are feelings that we refer to colloquially as "normal."<br />
<br />
If you're anything like me, taking home <i>everything</i> will be self-defeating. You'll be staring at that huge pile of paperwork and just get to a point where you're like, "<a href="http://memegenerator.co/instance/31762157" target="_blank">screw this</a>."<br />
<br />
I mean, there <i>will</i> be nights where a marathon grading session can't be avoided. It happens. But most nights, be realistic. Take home only what you can carry in your hands, and get that much done.<br />
<br />
<b></b>
<b></b>
<b> </b><br />
<b>5. Exercise.</b><br />
<br />
I wish I'd done more of this during my internship. Actually, I wish I'd done more of this my 20's entirely.<br />
<br />
I love to run, and I'm currently training for the Chicago Marathon (because I'm out of my mind). I started doing long distance races a couple of years ago and have been doing a <i>lot</i> of running. I've lost weight, lowered my blood pressure,
and lowered my cholesterol.<br />
<br />
Exercising regularly can help you be more energetic and feel more
positive during your pre-service experience or your first year of
teaching. It also serves as a way to keep you in an established routine,
which can be tremendously helpful in organizing the rest of your time.<br />
<br />
Also, it's pretty freaking cool when your colleagues start to notice you've been losing weight. The compliments can be very uplifting, especially when you're in the doldrums of late winter/early spring.<br />
<br />
How do you best like to exercise? Running? Biking? Swimming?
Rollerblading? Basketball? Tennis? Tae Kwon Do? Find out your preferred
method of staying active, and set aside time to do it at least 3-4 days a
week. Some teachers prefer to work out before school, some prefer to
work out after.<br />
<br />
Find out what works best for you, do it, and stay active. (Even better,
see if any of your colleagues or anyone from your cohort will exercise
with you!)<br />
<br />
<br />
<b>6. Eat healthy (or eat, period).</b><br />
<br />
I lived by myself during my internship, and I was God-awful at keeping
my place stocked with food. I'd skip breakfast. I ordered out a lot. As I
like to tell people, I was sustained by <a href="http://www.gumbyspizza.com/" target="_blank">Pokey Stix</a> and <a href="http://www.insomniacookies.com/" target="_blank">Insomnia Cookies</a> during my internship year. Thank goodness my wife started making me eat healthier when we moved in together, or I'd probably weigh eleventy billion pounds by now. Y'know, give or take.<br />
<br />
You'll definitely be busy this year. You may think you won't always have
time to cook, let alone eat healthy. It's even more difficult if you
happen to be living alone. But, there are ways to make it happen.<br />
<br />
Plan out all of your meals for a week (or even two) ahead of
time, put together a grocery list, and shop for everything at once. Do
this on the weekend. For healthy recipes or recipes that don't take very
long to put together, I highly recommend <a href="http://www.cleaneatingmag.com/" target="_blank">CleanEatingMag</a> and <a href="http://www.eatingwell.com/" target="_blank">EatingWell</a>.<br />
<br />
Bringing leftovers from last night's dinner for lunch is another great way to go with meal preparation. <br />
<br />
If you're <i>really</i> lucky (which I am) and have access to ample refrigerator and cupboard space at your school (which I do), then keep your own supply of breakfast and lunch items around. I keep milk and cereal for breakfast, and stuff to make sandwiches or salads for lunch. This way, I don't have to mess with making meals at home; I can just do it when I get to school.<br />
<br />
Teaching makes you hungry. Eat three meals a day. Have some snacks
handy, too. Just be sure to eat healthy and prepare your own meals
whenever you can. (You'll even spend less money!)<br />
<br />
<br />
<b>7. GO. TO. BED.</b><br />
<br />
You need to sleep. Seriously. Go bed at the same time, every night. Get at least 7-8 hours.<br />
<br />
You won't be able to function at your best if you're up
past midnight worrying about lesson plans or grading papers. Yes, the
grading needs to get done, but you have to balance that with your
health. (This is another good reason to obey the wisdom of #4, above.)<br />
<br />
Pick a bedtime and stick to it consistently, with little or no exception.<br />
<br />
You might actually <i>save</i> time by doing this. I got myself
into a very messy pattern of staying up very late working on plans or
grading, getting up at 5 AM (sometimes as early as 4 AM) the next morning, going to school all day,
coming home and immediately napping for three hours on the couch out of
sheer exhaustion. Sometimes I wouldn't wake back up until 8 or 9 PM, and
I'd still have work to do (because I was stupid and brought everything home and didn't know how to turn off the "teacher mode" switch, all while not exercising or eating healthy and thus violating pretty much every piece of advice I'm giving you). If I'd stayed awake, I'd probably have
gotten all of my work done and then had the rest of the evening to
myself.<br />
<br />
Get into a regular bedtime routine and avoid naps if at all possible. Use daylight to work, nighttime to rest.<b> </b>(I mean, not <i>literal</i> daylight, because the days are really short in the winter, but... well, you know what I'm getting at.)<br />
<br />
<br />
<b>8. But before you go to bed, be sure to unwind first.</b><br />
<br />
Going to bed at a regular bedtime isn't <i>quite</i> as effective if you're working frantically on grading or planning lessons up until the last minute. You're probably more likely to have fits of insomnia if you do this.<br />
<br />
Case in point: Just the other night, I was up working on a lesson for the upcoming school year. I worked until it was pretty late (hey, it's summer, the rules are different) and then went straight to bed.<br />
<br />
That was a mistake; I lay awake half the night thinking about the lesson I was planning on, because it was the last thing I was doing immediately before going to bed.<br />
<br />
Be sure to take at least an hour or so before bedtime to unwind and relax. Read a book. Play some video games. Netflix a TV show. A few months ago, my wife and I started watching <i>Doctor Who</i>, and we'll probably be continuing that as an evening ritual when the school year starts. It gives you time to relax before bed, and it gives your brain a chance to separate itself from work mode and sleep mode.<br />
<br />
Go to bed at a regular time, but don't go to bed without distancing yourself from your work first.<b> </b><br />
<br />
<br />
<b>9. Choose one night as your "FUN ONLY" zone.</b><br />
<br />
Just as important as learning how to turn off the "teacher switch" is
taking some time during the week to have fun. You need one night during the week to be your inviolable holy sanctuary of "me time."<br />
<br />
For me, this night was Friday. Most people pick Friday as this day, but some people prefer to use Friday to work ahead and enjoy the rest of the weekend (see #10, below). You may feel overwhelmed with work, even when the weekend is just kicking off. <i>Give yourself permission to take a break</i>. If you don't stop to enjoy some free time, you're going to burn out in a hurry.<br />
<br />
Designate one night as a "fun only" time. Go to a movie, go on a date, go have drinks with friends (note: you should probably <i>only</i> do this at the end of the week), whatever you want. Just make sure
you're taking time every week to do something fun.<br />
<br />
But don't let yourself get <i>too</i> out of control before the weekend, because...<b> </b><br />
<br />
<br />
<b>10. "Early in the weekend" is a great time to get some work done.</b><br />
<br />
Yes, I said to leave school at school whenever possible. But as one week ends, you'll have an entirely new week to plan for.<br />
<b> </b><br />
By all means, go out on Friday night if you want to. Sleep in on Saturday if that's what you need to recharge. You've earned it.<br />
<br />
But don't wait until Sunday evening to <i>start</i> planning for the week
ahead. You'll save yourself a lot of needless stress and worry by taking
a couple of hours earlier in the weekend to get some work done.<br />
<br />
When I say "early in the weekend," I'm being a bit broad. When I posted about this last year, I was specifically talking about Saturday morning. However, a few teachers mentioned to me that they prefer to get extra work done on Friday nights and <i>then</i> enjoy the rest of the weekend. One teacher commented that a colleague would put in extra time on <i>Thursday</i> nights, so she could enjoy Friday night, Saturday, and Sunday. So really, if you're the forward-planning type, you can get started on "next week" before "this week" is over. <br />
<br />
I never followed this advice during my internship. (Really, did I follow <i>any</i> of this advice? Damn you, past Jeff.)<br />
<br />
I was <i>masterful</i> at the art of procrastination; but having all of that
work hanging over my head each weekend not only detracted from my
ability to enjoy my free time, it also had a cumulative effect throughout the
year. I got to the point in March and April of my internship year when I
was having a hard time falling asleep on <i>Saturday</i> nights, let alone Sunday nights. I would actually be curled into a fetal position around noon on Saturday. That's no way to spend your weekend.<br />
<br />
You might not want to do it, but chances are you'll be able to enjoy
your weekend a little more if you're proactive during your weekend time (or if you're <i>really</i> ambitious, on Thursday night).<br />
<b> </b><br />
<br />
<b>11. Year 2 will be better.</b><br />
<br />
Ask just about anyone who is in the teaching profession. <b> </b>When
you've gone around the block once, things start to click. You feel more
confident about your teaching during your second year because you have a sense of "I've done
this before." You have something you didn't have in your first year:
experience.<br />
<br />
Your experience will inform your practice as
you take time to reflect and make adjustments. You'll have a better idea
of what works and what doesn't. You'll find ways to manage your
classroom and engage your students that are better, more efficient.<br />
<br />
(By
the way, you can do yourself a great service during your first year by <i>making notes</i>
of what works well and what needs changed. Take time to reflect on your
practice as often as you can. Future you will thank past you for it. In fact, this is another great reason to <i>start a blog</i>.)<br />
<br />
If the idea that your second year of teaching will be better than your
first gets you through, then by all means hold onto it. I can tell you
that the third year is even better than the second
year. I just finished my fourth year, and it was by far and away my best year ever. I'm super-excited for my fifth year; in fact, I don't think I've <i>ever</i> been as excited to start a new school year as I am now.<br />
<br />
It gets better.<br />
<br />
If teaching is your passion, that's all the truth you need this year.<br />
<br />
Good luck. Be awesome.<br />
<br />
-----<br />
<br />
Once again, veteran teachers, if you would like to add anything, please comment below!<br />
<br />
Second-year teachers, if you have any advice, I'd love to hear from you too. :-)<br />
<br />
First-year teachers and/or interns, join the conversation!<br />
<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com4tag:blogger.com,1999:blog-3248548197809593245.post-54492364250257128572013-08-08T10:04:00.001-05:002013-08-08T10:04:05.621-05:00Mathspotting (Because Math Hides In Plain Sight Like a Ninja)One of the coolest things about doing math for a living is having a higher sensitivity to its presence in the world during day-to-day activities. For me, this seems to be particularly true right before the school year when my brain is constantly in planning mode. So I'm, like, on HIGH MATH ALERT.<br />
<br />
I went to the beach with my dog yesterday morning, and noticed several sets of tire tracks in the sand. There are many different types of tire patterns, of course, but this particular set caught my eye (so I took a photo and tweeted it):<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ2KLC-mZ2pXHxLl2UxToLc0gd2mNngrKy8MzqO5P0wzKDqmBoC3kZQhREURPOqoFYlfj00CpsItHv4nLgSwj28evuwP-MVnEf9uiHIxAT52ww7kdHnR_oDcr0AtcfxDk80KVqUO1xAqk4/s1600/Screen+Shot+2013-08-08+at+8.14.49+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ2KLC-mZ2pXHxLl2UxToLc0gd2mNngrKy8MzqO5P0wzKDqmBoC3kZQhREURPOqoFYlfj00CpsItHv4nLgSwj28evuwP-MVnEf9uiHIxAT52ww7kdHnR_oDcr0AtcfxDk80KVqUO1xAqk4/s1600/Screen+Shot+2013-08-08+at+8.14.49+AM.png" /></a></div>
Of course, I'm not completely sure that these are <i>exactly</i> trig patterns, but... but... close enough, right? RIGHT?<br />
<br />
In any case, it had me wondering about the application (if there actually is any) of periodic functions in designing certain types of tire treads. I don't really know anything about how tires are designed, so take it for what it's worth. But it's cool to think about; I mean, if I could legitimately tell a student that trigonometry is what keeps them from hydroplaning in a downpour, that would be <i>awesome</i>. I just don't know if that's actually true or not. *shrug*<br />
<br />
In the evening, my wife and I were walking our dog and made a quick stop at the grocery store. As I was waiting outside with the dog, I found myself staring at this sign in front of me and wondering mathy things:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnHXrzSrGna0j6de30v4_OCLDSO2c9KFCJ0rGFjvQOlKTjuepUqc5_D9GEhkudq5yzIq-h5zw1EStIs8ECuiDGG_GphXTf0WPVSq3o7U7hVUJ9RIRVI5nSz-Ay7V0758rR4CN31Rp9rTYG/s1600/Screen+Shot+2013-08-08+at+9.48.04+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnHXrzSrGna0j6de30v4_OCLDSO2c9KFCJ0rGFjvQOlKTjuepUqc5_D9GEhkudq5yzIq-h5zw1EStIs8ECuiDGG_GphXTf0WPVSq3o7U7hVUJ9RIRVI5nSz-Ay7V0758rR4CN31Rp9rTYG/s1600/Screen+Shot+2013-08-08+at+9.48.04+AM.png" /></a></div>
I actually look at signs like this and think about symmetry problems all the time. Like, probably an unhealthy amount. If you see me staring at a sign, chances are I'm probably thinking about symmetry. I try not to do it while driving.<br />
<br />
Anyway, when I posted this on Twitter, one of the comments I got was: "What kind of symmetry? Even or odd?" Which is <i>exactly </i>the kind of question I was hoping to see. If I posed this problem to my students (and I may very well do that), I would love for this issue to arise. The "N"s certainly have odd symmetry, and I never did specify any particular type of symmetry. So, we'd have to include the "N"s in our answer, yes?<br />
<br />
So those are just a couple of math nuggets that I spotted yesterday. Maybe I should post more mathy pictures on Twitter and start hashtagging them with #mathspotting or something. Feel free to join in!<br />
<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com3tag:blogger.com,1999:blog-3248548197809593245.post-3440266838822777872013-08-05T23:44:00.002-05:002013-08-05T23:44:41.565-05:00"When Am I Ever Going to Use This?" ...Sometimes I Don't Know the AnswerFor some reason, the question of "when am I ever going to use this in real life?" seems to pop up at a disproportionately higher rate in math class than in any other subject. I'm willing to bet that's the case.<br />
<br />
Do I have any scholarly research or statistics to back up this claim? No.<br />
<br />
But I <i>did</i> go to Google and type in the phrase, "when am I ever," to see what popped up:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpibPHmfJVAV_sepjNCxuxsiQT1kcquvMuV7waSxZvTWd3B5gDwANXPgvGsM269wx1v1yTbSHsKCXvcRkYZzQ78E6OgWS7qOW9_F1IK3E6X5JzF7ELWExGuHnBVsSJEVUBhIYGD-uVzaOn/s1600/Screen+Shot+2013-08-05+at+10.01.04+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="74" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpibPHmfJVAV_sepjNCxuxsiQT1kcquvMuV7waSxZvTWd3B5gDwANXPgvGsM269wx1v1yTbSHsKCXvcRkYZzQ78E6OgWS7qOW9_F1IK3E6X5JzF7ELWExGuHnBVsSJEVUBhIYGD-uVzaOn/s320/Screen+Shot+2013-08-05+at+10.01.04+PM.png" width="320" /></a></div>
See? Algebra and Calculus! Obvious proof that this question is asked more in math class than in any other class! And if you aren't convinced by this, then... uh... um... well, then you probably think critically about your info sources and have good judgment.<br />
<br />
At any rate, the new school year is around the corner. I've enjoyed the time away from the classroom and have spent many hours mapping curriculum, trying to keep up with the happenings in the <a href="http://samjshah.com/2013/07/19/exploring-the-mathtwitterblogosphere/" target="_blank">MTBoS </a>and <a href="http://brennemath.blogspot.com/2013/07/battleship-graphing-equations-of-circles.html" target="_blank">preparing some new tasks</a> to try out this year.<br />
<br />
As I was reflecting on my first four years of teaching and looking ahead to year five, I kept thinking about what I'm going to do when, inevitably, the question is asked:<br />
<br />
<i>"When am I ever going to use this kind of math in real life?"</i><br />
<br />
This question nearly always evokes some kind of emotional reaction from me. One of two types, in fact:<br />
<br />
(1) UNADULTERATED, ABSOLUTE JOY, because I have an answer to the question
that is totally satisfactory, underscores the relevance of the current
mathematical topic, lets me talk about math (which is super-cool because I <i>love</i> talking about math) and <i>helps the kid to see just how motherfreaking awesome math is</i>,<br />
<br />
or<br />
<br />
(2) MURDEROUS RAGE that someone, who's half my age, who hasn't even <i>learned</i> as much math as I've <i>forgotten</i>, would have the <i>impudence</i> to ask me that question<i>. </i>Not just to ask me that question, but to ask me that question when I have <i>no idea whatsoever how to answer in a way that isn't complete bullcrap.</i><br />
<br />
Okay, I don't <i>actually</i>
get mad at students for asking me that question. Or any question. Not ever. I like having curious students. And a job.<br />
<br />
I do
try my best to be prepared to answer the question of "when am I going to use this?" for any mathematical
topic that comes up in my classroom. But sometimes I do feel annoyed
when I don't really have a good answer. Not annoyed at the student (not much, anyway), but more at myself for not being prepared with a brilliant, insightful response. After all, <i>I'm</i> the math teacher, right? I should know, in great detail, when the hell someone would ever use an inverse tangent function when they get out into the real world. I should be able to spell out the <i>exact</i> situation in which one would need to know everything there is to know about the <a href="http://mathworld.wolfram.com/LatusRectum.html" target="_blank">latus rectum</a>. (Tee-hee.)<br />
<br />
But I don't always know the answer. When that happens -- depending on my mood/how busy I am/what's happening in class/how many cups of coffee I've consumed in the past five minutes -- I tend to go with one of the following responses (I don't recommend using any of these):<br />
<ul>
<li>"That's a great question." *flees without saying another word*</li>
<li>"I can't tell you that; it would ruin the surprise!"</li>
<li>"'Real life?' Math <i>is</i> real life, son."</li>
<li>"You know, I find that the best answers in life are the ones we find for ourselves."</li>
<li>"What're you talking about? <i>I</i> use it all the time!" ("But you're a math teacher," the student replies. "Yep, that's why I use it all the time!" I reply back.) </li>
<li>"Uh, come see me after class and I'll be happy to talk to you more about your question." (Nobody has ever taken me up on this.)</li>
</ul>
<br />
I feel terrible when I don't have an answer for that question right away. There are times when it seems like it would be easiest just to say, "you know, there's a pretty good chance that you're not actually going to use this; but hey, gotta know it to pass, right?"<br />
<br />
I mean, I've <i>actually</i> uttered those words to a student once or twice. I'm not proud of it. At the time, I felt like I was just being straight with the kid(s) who asked. I guess I figured that kids appreciate honesty and have a pretty good nose for B.S. But when I think about it, I realize that what I really did was cheat those students out of a great learning opportunity. I cheated myself as well.<br />
<br />
I'm a math teacher, yes, but I'm also a math <i>learner</i>. A <i>lifelong</i> math learner. I shouldn't be ashamed or annoyed when I don't know exactly where or when stuff like hyperbolas or the mean value theorem are used in real life. Instead, I should be seeing an opportunity to learn something new. I should be excited that I've discovered something new to learn about a topic I absolutely <i>love</i>. I should be, like, <i>absolutely jacked that there's stuff about math that I don't know but can find out about for myself. </i>I mean, that's what I'd want my students to do, right?<br />
<br />
So this year, I'm going to start using this response (or something similar, still a work in progress):<br />
<ul>
<li>"You know what? I'm not quite sure. I know there's a use for [<i>insert mathematical concept here</i>], but I'm still trying to figure that out. I'll try to do some research on it after class. Maybe you could also look it up, and let me know what you find. That would be really helpful."</li>
</ul>
<br />
I'm a math teacher; I shouldn't be shrinking away from that question when a student comes asking. I should be full-on body tackling that thing like it's a quarterback's blind side.<br />
<br />
I do think we math teachers <i>try</i> to know where the stuff we teach can be used in the world beyond school; but we won't always know. When we don't know, we need to find out. We need to include our students in the process of finding out, because they asked the question in the first place.<br />
<br />
We can't always know, but we <i>can</i> always care. We can always care enough to try and find out. We can always care enough to try and do better.<br />
<br />
So this year, I'll try and do better.<br />
<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com4tag:blogger.com,1999:blog-3248548197809593245.post-76017450418494270222013-07-28T14:33:00.001-05:002013-08-06T12:55:12.629-05:00BATTLESHIP! - Graphing Equations of CirclesI've been dying to incorporate more PrBL tasks into my classroom. For the past couple of years, our math team spent a huge deal of time and energy on a complete overhaul of our four-year math curriculum in order to more strongly align it with ACT College Readiness Standards. It was certainly a worthwhile endeavor; I'm very proud of what our awesome math team has accomplished, and I think our students will greatly benefit from what we've done so far.<br />
<br />
At the same time, this pretty much meant I had zero time to work on any PrBL stuff, especially with moving from teaching Geometry to teaching Pre-Calculus at the same time. However, our project was finally completed this past spring, so I have been happily spending the summer working on PrBL-related curriculum mapping for my Pre-Calculus and Advanced Pre-Calculus classes.<br />
<br />
(Yes, I just said "happily" and "curriculum mapping" in the same sentence.) <br />
<br />
Below is one PrBL task that I've been working on for a graphing unit this school year. I think (and hope) the students will have fun with it; it's not particularly all that "real-worldy," and it definitely needs refinement, but I gotta start somewhere. Of course, as with anything I haven't tried in class yet, it's a work in progress.<br />
<br />
This task involves understanding and graphing equations of circles. I call it: <b>BATTLESHIP!</b><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOAFQDU1QitRRKUCzqwtXB0MwigKopw7YUjZnHEfLPzM3luIYYyW50-jwIgxCJxIhSWz5TIePu3fZ1_W6RHbWLm80H__bgcRj2ceC4nUvFzk7QqM7OFSZH4H3rJ3lUcEmeNvelIvzXwnI4/s1600/battleship.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="159" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOAFQDU1QitRRKUCzqwtXB0MwigKopw7YUjZnHEfLPzM3luIYYyW50-jwIgxCJxIhSWz5TIePu3fZ1_W6RHbWLm80H__bgcRj2ceC4nUvFzk7QqM7OFSZH4H3rJ3lUcEmeNvelIvzXwnI4/s320/battleship.jpg" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: xx-small;">(Although the task is not quite the same as the classic board game.)</span></div>
<br />
<b>The Scenario: </b><i>You are the commander of a mighty naval fleet in the middle of international waters. The enemy has developed a new type of submarine known as a Hyperbolic Invisibility/Deep Dive ENgine, or a H.I.D.D.EN. submarine.</i><br />
<i><br /></i>
<i>The enemy's H.I.D.D.EN. submarines are capable of avoiding nearly all types of radar detection. In fact, you are only able to determine the distance a H.I.D.D.EN. submarine is from any of your naval stations.</i><br />
<i><br /></i>
<i>Your task is to devise a way to pinpoint the exact location of a H.I.D.D.EN. submarine. Succeed, and your forces will be able to destroy the enemy fleet. Fail, and you're doomed. DOOMED!</i><br />
<br />
(If you couldn't tell, I have an affinity for silly acronyms.)<br />
<br />
<br />
<b>The Entry Event:</b> Before things really kick off, I'll give the students a few warm-up problems to assess and activate their prior knowledge. Students will need to know the parts of a circle (particularly the radius and the center), and will also need to be able to re-write a two-variable equation (i.e. solve for <i>y</i> in terms of <i>x</i>). The latter will be important for graphing circles on most graphing utilities.<br />
<br />
To introduce the problem, I'll present the following situation to students on <a href="http://activeprompt.herokuapp.com/" target="_blank">Activeprompt</a>:<br />
<br />
<i>"You are the commander of a naval station, shown here on the grid. An enemy submarine is approaching.</i><br />
<br />
<i>The submarine has a cloaking device that hides its exact location from your radar system. However, you are still able determine how far away the submarine is from the station.</i><br />
<br />
<i>The submarine is 5 miles away from the station. Where is it?" </i> <br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_0BJIc3X52JPZP1mHcytmCghcJgyqZnoi6vYDRL19T2VjfSYNW_9CIC63GjFzaUv1tlOCOXaAsMgH_4iuRupSwjOMmllZRLePlb7EatKJNxeBpjUWSAWu-HAqyA3XNFHCh-ah2tlNHyY8/s1600/Screen+Shot+2013-07-27+at+6.44.59+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="369" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_0BJIc3X52JPZP1mHcytmCghcJgyqZnoi6vYDRL19T2VjfSYNW_9CIC63GjFzaUv1tlOCOXaAsMgH_4iuRupSwjOMmllZRLePlb7EatKJNxeBpjUWSAWu-HAqyA3XNFHCh-ah2tlNHyY8/s640/Screen+Shot+2013-07-27+at+6.44.59+PM.png" width="640" /></a></div>
<div style="text-align: center;">
<span style="font-size: xx-small;">(I could make things more interesting by removing the axes and labels, but I want to steer the students in a certain direction here.)</span></div>
<br />
I posted this prompt on <a href="https://twitter.com/brennemania" target="_blank">Twitter</a>, and a couple of my friends immediately pointed out that they couldn't answer the question because they didn't know which direction the submarine was from the station. This is true, and in many ways is actually the point of this prompt; I suppose I should be more clear that I want students to <i>guess</i> where the submarine <i>might</i> be, and that I'm not necessarily looking for "the correct answer" at this stage.<br />
<br />
Still, I had several responses to the prompt and ended up with something I would hope to see in class:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyBgOJo8ADJ83oxLZX2qPzGDnIqrSPoL8bk2Tg6efHL2t7Qjjos_qIQrJ1HExqi95P7zjdLeHywi1WzWoKTxA7WykLx4YoZsDwyeFVqMc0fUhctemnvLFTrm2CmRcLqUEfe7O4qAoSQq7s/s1600/Screen+Shot+2013-07-27+at+10.09.26+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="372" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyBgOJo8ADJ83oxLZX2qPzGDnIqrSPoL8bk2Tg6efHL2t7Qjjos_qIQrJ1HExqi95P7zjdLeHywi1WzWoKTxA7WykLx4YoZsDwyeFVqMc0fUhctemnvLFTrm2CmRcLqUEfe7O4qAoSQq7s/s640/Screen+Shot+2013-07-27+at+10.09.26+PM.png" width="640" /></a></div>
<div style="text-align: center;">
<span style="font-size: xx-small;">(Interesting, isn't it?)</span></div>
<br />
Hopefully, students will take one look at this picture and notice the pattern: there <i>appears</i> to be a circle forming around the station. At this point, students can take some time to think about further questions: <i>Why</i> is there a circle? What does this circle mean? What can we figure out about this circle? What does this circle have to do with finding the submarine? <br />
<br />
After discussion, the hope is that students would come to the following conclusions:<br />
<ul>
<li><i>The circle represents all of the possible locations of the submarine, based on the information we have.</i></li>
<li><i>We have no way to determine the exact location of the submarine <u>with our current information</u>.</i></li>
</ul>
That second statement is critical. The key to solving the problem lies in the realization that more information is needed.<br />
<br />
<br />
<b>Need-to-Knows & Scaffolding:</b> While I'm sure that my students will surprise me (students have a habit of doing that), the need-to-know that should be immediately apparent is: <b><i>"How do we locate the submarine?" </i></b>In fact, we begin the process of answering this question with the entry event.<br />
<br />
Again, one of the key realizations from the entry event is that all of the possible locations of the submarine are represented by a circle, radius 5, with the naval station as its center:<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-9agKAF_O24dytC4z9YOUk33omu53QbtOznrvDzIVcto8kxyknqfMa-pkGtC0DBjDi3vkIeU7DfgEMLzT_Oaf1gbusYE9mbDsIi1pugXkhC4N6A5exxrnY3P2_3LShSQ-pQUePKudEoGU/s1600/Screen+Shot+2013-07-28+at+11.56.48+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-9agKAF_O24dytC4z9YOUk33omu53QbtOznrvDzIVcto8kxyknqfMa-pkGtC0DBjDi3vkIeU7DfgEMLzT_Oaf1gbusYE9mbDsIi1pugXkhC4N6A5exxrnY3P2_3LShSQ-pQUePKudEoGU/s320/Screen+Shot+2013-07-28+at+11.56.48+AM.png" width="320" /></a></div>
<br />
A good follow-up question would be, <i><b>"How do we narrow down the number of possible locations?"</b></i> The answer may or may not be readily apparent. I'd encourage students to think outside the box -- or perhaps, more appropriately, "think outside the circle."<br />
<br />
Because we could narrow down the number of possible locations if we had a <i>second naval station</i>. Say, at coordinates (7, -8). And it detects the submarine at a distance of 7 miles.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvkHAYhMy_pBGOpOm7PGXiR8vQXfGkjEpEj40rzUp4Vytbk-zWVweaqnDO4AxO4RxLKXSBkrw8Q7HBEsPLEqCCquwxUGOiqCmqcUKvOIeOICtz2e8HoX7lBFfsK1Eo7_w8tt06rc73DYvJ/s1600/Screen+Shot+2013-07-28+at+12.30.52+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvkHAYhMy_pBGOpOm7PGXiR8vQXfGkjEpEj40rzUp4Vytbk-zWVweaqnDO4AxO4RxLKXSBkrw8Q7HBEsPLEqCCquwxUGOiqCmqcUKvOIeOICtz2e8HoX7lBFfsK1Eo7_w8tt06rc73DYvJ/s640/Screen+Shot+2013-07-28+at+12.30.52+PM.png" width="530" /></a></div>
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Aha! Just like that, we've narrowed our possible locations down to two; namely, the two points (1 and 2) where the circles intersect each other. (It certainly wouldn't hurt to have the students explain why these are the only two places the submarine could be.)<br />
<br />
From here, it probably won't be a huge leap for the students to realize that adding a <i>third</i> naval station will narrow our choices down to just one. We'll get back to that in a moment.<br />
<br />
A critical issue arises from this new picture: while Point 2 clearly appears to be located at the coordinates (7, -1), it's much less clear what the coordinates of Point 1 are. This should lead to another question: <i><b>"How do we accurately determine the coordinates of the point(s) where the circles intersect?"</b></i><br />
<br />
Now, this part of the task is a bit murky for me. It's not all that difficult to come up with a good estimate of Point 1's coordinates using Geometer's Sketchpad, but the point of the task is for students to work with and understand equations of circles. To this end, I want students to be working with a graphing utility (e.g. TI-83/84) as we address this question. So, yeah... if anyone has a good suggestion for how to make sure it steers in that direction, I'm all ears!<br />
<br />
In any case, turning to our graphing calculators should bring up the question: <i><b>"How do we graph circles?"</b></i> The best way to do this with our graphing calculators (or an online tool like <a href="https://www.desmos.com/calculator" target="_blank">Desmos</a>) would be to input an equation. That, of course, leads to: <i><b>"What's the equation for a circle?"</b></i> <br />
<br />
At this point, appropriate scaffolding activities and workshops could be used to help students understand how to determine the equation of a circle, given the center and the radius. I'd probably also give students a few practice problems to give them some exercise in this skill. When using a graphing calculator like a TI-83 or TI-84, students would also need to know how to re-write their circle equations for <i>y</i> in terms of <i>x</i> so they can actually enter them. (This would be one advantage of using Desmos over a graphing calculator; such a conversion isn't necessary. On the other hand, re-writing equations would also be a great chance to talk about issues such as positive and negative roots, for instance.)<br />
<br />
Since I don't have the proper software readily available for getting some clear TI-83 screenshots, here are the two circles graphed on Desmos:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4Zwq1DvbeJ5oD1mMf0xAdErEXVNTxR9mQBUH4dcnBwdJFmOdVshagZYjlCaeYc-y1iduJsmLsidnqVWtKxa6S6_MT6yY913xS8_0eDNYMRdrCB4wzxXG5BHOf_qwBGbN-o36XhfFKNd9J/s1600/Screen+Shot+2013-07-28+at+1.28.46+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4Zwq1DvbeJ5oD1mMf0xAdErEXVNTxR9mQBUH4dcnBwdJFmOdVshagZYjlCaeYc-y1iduJsmLsidnqVWtKxa6S6_MT6yY913xS8_0eDNYMRdrCB4wzxXG5BHOf_qwBGbN-o36XhfFKNd9J/s640/Screen+Shot+2013-07-28+at+1.28.46+PM.png" width="640" /></a></div>
<br />
On a TI graphing calculator, students could use a combination of ZOOM and TRACE to estimate the coordinates of Point 1. CALC -> INTERSECT would also be a good option. On Desmos, we can just click on the intersection point to get an estimate of the coordinates:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3kG6UB1cPLv3fuIRLeAo4njDfekW1g-hJAPDvlD96sGESD6IJcMAa5przxxAYKcHy7CZgt3YkwqTgroOxTV-5z9s9piXpeCA6TLa0jbWL7-P0v26sPyvCaI8qMN8oz7a8SqwvA1hd7wP5/s1600/Screen+Shot+2013-07-28+at+1.50.03+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3kG6UB1cPLv3fuIRLeAo4njDfekW1g-hJAPDvlD96sGESD6IJcMAa5przxxAYKcHy7CZgt3YkwqTgroOxTV-5z9s9piXpeCA6TLa0jbWL7-P0v26sPyvCaI8qMN8oz7a8SqwvA1hd7wP5/s320/Screen+Shot+2013-07-28+at+1.50.03+PM.png" width="227" /></a></div>
If we want greater accuracy, we can zoom in really close: <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcCbmaYSn2pbGxeaNS6dm5UwbDaet-oakcff8Hb8vjRHeJ2jWiiB-dKfL9YB_ZuZZQ7xKIivsBLaOtKlX3vELSrn_HeQ_TSzm5Pi5rG_VmRMvJxpmOlMubpPgg0ApX8cFOL6MqDBjR9Rfz/s1600/Screen+Shot+2013-07-28+at+1.50.35+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcCbmaYSn2pbGxeaNS6dm5UwbDaet-oakcff8Hb8vjRHeJ2jWiiB-dKfL9YB_ZuZZQ7xKIivsBLaOtKlX3vELSrn_HeQ_TSzm5Pi5rG_VmRMvJxpmOlMubpPgg0ApX8cFOL6MqDBjR9Rfz/s320/Screen+Shot+2013-07-28+at+1.50.35+PM.png" width="320" /></a></div>
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<br />
Using CALC -> INTERSECT on my TI-83 yielded an estimate of (3.4461538, -1.969231), so very similar results. If we rounded to the nearest hundredth, we can pretty solidly estimate the coordinates of Point 1 to be (3.45, -1.97). (It might be interesting to have students estimate the coordinates of Point 1 <i>prior</i> to using their graphing utilities to see how close they came by just "eyeballing" it.)<br />
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Of course, we said much earlier that we need <i>three</i> stations to determine where the submarine is. We could introduce the third station much earlier in the problem, or we could hold off until now to introduce it.<br />
<br />
So, let's say the third station is located at (-5, 4) and detects the submarine at a range of 13 miles. Students determine the equation of the circle with this center and radius, enter it into their graphing utility, and <i>voila</i>:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbs87qX59sfHsMkMVwCeuV8DSmPtWjNMvSj3UsMRi_eqPCv_S0DA_KD5Wo8q_7aaMPLT-5vEhwKGCMl-lBFxXrDVFcSrce0EZE2HdHmZ2Jn0RG2O3_OuUJeJVqAoCU1zYn9U8kPuNsWw2H/s1600/Screen+Shot+2013-07-28+at+1.56.14+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbs87qX59sfHsMkMVwCeuV8DSmPtWjNMvSj3UsMRi_eqPCv_S0DA_KD5Wo8q_7aaMPLT-5vEhwKGCMl-lBFxXrDVFcSrce0EZE2HdHmZ2Jn0RG2O3_OuUJeJVqAoCU1zYn9U8kPuNsWw2H/s640/Screen+Shot+2013-07-28+at+1.56.14+PM.png" width="640" /></a></div>
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So our enemy submarine is located at coordinates (7, -1). Huzzah!<br />
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<br />
<b>Applying the Learning:</b> Now, I wouldn't have gone through the whole business of figuring out how to estimate coordinates using a graphing utility if the solution was always going to be as simple as (7, -1). For something more challenging that <i>definitely</i> requires the assistance of a graphing utility, let's say we have the following information:<br />
<ul>
<li>Naval Station A is located at (-16.47, -3.53). It detects an enemy submarine at a distance of 12.31 miles.</li>
<li>Naval Station B is located at (5.68, -3.74). It detects an enemy submarine at a distance of 13.97 miles.</li>
<li>Naval Station C is located at (5.43, 5.68). It detects an enemy submarine at a distance of 11.96 miles.</li>
</ul>
This takes a bit more work, and the answers will probably vary slightly. This might also be a good opportunity for students to debate how to get the "best" answer to this problem, since we want to be as accurate as possible when tracking down the enemy submarine. <br />
<br />
<br />
<b>Would You Like to Play a Game?:</b> For something <i>really</i> fun at the end of this problem, we would turn our scenario into a war game. I would break the students up into teams of two or three; each team gets one H.I.D.D.EN. submarine and three naval stations. Teams get to place their submarine and naval stations at whatever coordinates they choose (within certain borders, of course).<br />
<br />
After all submarines and stations are placed, I provide each team with information about how far away each enemy submarine is located from their stations. (This adds a layer of complexity to the original problem scenario, as teams now have information about multiple submarines and they have to mix & match circles in order to pinpoint them all.) The teams then race against each other to try and be the first to locate and destroy the other submarines. Winning team gets riches and glory. Well, just glory. Not much glory.<br />
<br />
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<b>Final Thoughts:</b> In the end, I thought this task seemed like a fun way for students to learn about how to graph equations of circles and then apply that skill.<br />
<br />
Hopefully, when the students share out what they learned as a result of this problem, they'll be able to articulate a deep understanding of the relationship between circles, their equations, and their graphs. It'd also be cool if some of them see the connections between equations of circles and the Pythagorean Theorem or the Distance Formula. I certainly hope they end up finding the whole thing to be a worthwhile experience. <br />
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It's definitely not perfect, but I'm looking forward to trying it out and seeing how it goes.<br />
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<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com0tag:blogger.com,1999:blog-3248548197809593245.post-37379680337342821282013-07-16T15:39:00.000-05:002013-07-16T15:40:18.574-05:00Huh? What? Where?Yesterday morning, an e-mail found it's way to my inbox to alert me that <a href="http://crazedmummy.wordpress.com/" target="_blank">crazedmummy</a> left a comment on a previous entry of mine.<br />
<br />
The comment read, <i>"I nominated you, get back to work!"</i> and was accompanied by a healthy list of questions about me.<br />
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At first I was like, "HUH? WHAT? WHERE?" because I had no idea what I was being "nominated" for. I thought maybe there had been a secret gathering of blogging ninjas that decided to select other bloggers among those who have been inactive for months and leave cryptic messages on their blogs, telling them to get back to work. And if a blogger <i>didn't</i> get back to work, then the secret blogging ninjas would break into their house/apartment/cardboard box in the middle of the night while they slept and do something horrible, like tattoo the University of Michigan logo on their forehead. I mean, that's just awful. Like, death would be preferable.<br />
<br />
But then I did a bit of digging around, and found that I was really being nominated for a Liebster Award. It's a prestigious award that involves widespread fame and endless riches.<br />
<br />
Or maybe not.<br />
<br />
In reality, it's not really an "award," but rather a way for bloggers to recognize other bloggers that are either getting started or have less than 200 followers on their blog. (I'm <i>just</i> shy of this number at a hefty 6 followers total.) <br />
<br />
I'm not sure where it originated from. As far as I can tell, in order to accept the nomination for the Liebster Award, I must complete these tasks:<br />
<br />
<i>1. Link back to the blog that nominated you<br />
2. Nominate 5 blogs with fewer than 200 followers<br />
3. Answer the questions posted by your nominator<br />
4. Share 11 random facts about yourself<br />
5. Create 11 questions for your nominees<br />
6. Contact your nominees and let them know you nominated them</i><br />
<br />
Sounds like a great opportunity for me to get off my butt and resume blogging, thus proving to the rest of the world that I am, in fact, not at all dead.<br />
<br />
So, step one is to link back to the blog that nominated me. Did that at the top of this post, but hey, <a href="http://crazedmummy.wordpress.com/" target="_blank">here's crazedmummy's blog once more</a>.<br />
<br />
Step two, nominate 5 blogs with fewer than 200 followers. I don't really know how to check for that, but I don't think the blogging ninjas are gonna come after me for this one, so here's five blogs:<br />
<br />
<a href="http://jdevarona.wordpress.com/" target="_blank">The Problem Bank</a><br />
<br />
<a href="http://intersectpai.blogspot.com/" target="_blank">The Pai Intersect</a><br />
<br />
<a href="http://theeducationoffuturemathninjas.wordpress.com/" target="_blank">The Education of Future Math Ninjas</a><br />
<br />
<a href="http://mrwardteaches.wordpress.com/" target="_blank">Prime Factors</a><br />
<br />
<a href="http://writingtolearntoteach.wordpress.com/" target="_blank">Writing to Learn to Teach</a><br />
<br />
<br />
Step three, answer the questions that were posed to me. Here goes!<br />
<br />
<b>1. Why did you start this blog?</b><br />
Because all the cool kids were doing it! Actually, when I started this blog last summer, I was intending to use it as a place to share ideas for problem-based learning, to talk about ongoing issues in education, and to network with other educators who are active on the blogosphere. Things kinda ground to a halt as the school year got underway; I'm hoping to be more diligent in my blogging this coming school year.<br />
<br />
<b>2. What do you like most about your life?</b><br />
I'm happily married to my best friend in the world; we have a wonderful dog, a place to live, and food to eat; and, I love my job. There's not much that isn't to like!<br />
<br />
<b>3. What one thing would you improve about yourself?</b><br />
Well, I'm hoping to blog more and do a much better job of staying engaged with other teacher/bloggers this year than I did last year. I feel like I'm missing out on a lot!<br />
<b><br />4. What sort of vehicle do you drive to work?</b><br />
A 2000 Nissan Altima. Previously, it was a 2003 Dodge Neon, but in February I hit a deer and totaled it. Ah well, life goes on. (Except for the deer.)<br />
<br />
<b>5. How often do you cook?</b><br />
My wife and I split it pretty evenly. I actually rather enjoy cooking!<b> </b><br />
<b><br />6. What one show do you watch on TV ( if any)?</b><br />
I'm actually into a couple of shows right now: my wife and I have been working our way through Doctor Who (the 2005 and onward series), and this summer I've gotten into Breaking Bad.<b> </b><br />
<br />
<b>7. What non-school talent do you have?</b><br />
I sing with my wife in our church choir, and I write creatively from time to time. Those count, right?<br />
<br />
<b>8. What is your favorite exercise?</b><br />
Running! I'm currently training for the Chicago Marathon and I'm pretty excited for it.<b> </b><br />
<br />
<b>9. Where would you go for vacation if you had unlimited funds?</b><br />
I can't pick just one place, but since I have unlimited funds I supposed I could spring for a grand tour of various destinations. Among these would definitely be Hawaii, Australia, Tahiti, England, Ireland, Italy, and France.<b> </b><br />
<br />
<b>10. What one thing do other teachers do that you think is completely crazy (only one).</b><br />
Keep coming back every fall. Come on, we're <i>all</i> crazy for liking this job, aren't we?<br />
<br />
<b>11. If you were not a teacher, what else would you do?</b><br />
Oh, let's dream here. Maybe I'd go into radio or do public address announcing. Maybe I'd spend time learning how to use music software and compose soundtracks for video games. Maybe I'd write TV sitcoms. I dunno, those all sound fun. But I <i>love</i> teaching.<br />
<br />
Phew, this is a lot of work, but good for getting the blogging juices flowing. Alrighty, step four:<br />
<br />
<br />
<b>Eleven Random Facts About Me!</b><br />
<br />
1. I love many classic video game series, like Mario, Zelda, Mega Man, Final Fantasy, etc. But one series that I'm really into that you're far less likely to have heard of is Phoenix Wright: Ace Attorney. Great storylines, awesome characters, and sweet soundtracks.<br />
<br />
2. Speaking of video games, I'm totally into video game music (VGM) cover bands. Go onto Google and check out The One-Ups, The Megas, The Protomen, Descendants of Erdrick, The World is Square, and Bit Brigade for starters.<br />
<br />
3. A life ambition of mine is to visit all 50 states and all 7 continents with my wife. So far, we're at like, 6 states and 1 continent, or something. But we have future travel plans!<br />
<br />
4. My favorite pizza is Domino's, hands down. However, I do miss getting Pokey stix from Gumby's in East Lansing. I lived off of those during my year of student teaching (terrible idea, by the way).<br />
<br />
5. I once broke a window with a computer processor. Long story.<br />
<br />
6. There's a <a href="http://www.youtube.com/watch?v=k0KgkIMHilU" target="_blank">song about me</a> on YouTube that I got for donating to a Kickstarter.<br />
<br />
7. When I was attending Michigan State University for my undergrad, I worked for the freshman orientation program during the summer. This included giving campus tours daily; I became adept at walking backwards while spouting all kind of random facts and figures about the university.<br />
<br />
8. I also worked a a summer camp for a few years after I finished college; my favorite thing was to threaten the kids with liver and onions for dinner if they didn't manage to successfully shoot their archery targets. It was an excellent motivator; liver and onions were never served once.<br />
<br />
9. I took a job working as a night receptionist while at Michigan State purely because I was having major trouble with insomnia at the time. I figured if I was going to be awake all night, I might as well be getting paid for it. I worked a few semesters, including one summer. Working night reception during the summer was the <i>best</i> because there was pretty much nobody around, and the only people who came to the door had their own keys anyway. I'd take my TV and PS2 and just play video games all night long, <i>and get paid for it</i>. Sometimes I miss that.<br />
<br />
10. I'm deathly afraid of roller coasters. I think it stems from a trip to Disneyland when I was 3 years old. My parents took me on Space Mountain and it scared the life out of me. I've never been a fan of roller coasters and I don't think I ever will be.<br />
<br />
11. My favorite flavor of any kind just might be pumpkin. I haven't met anything pumpkin-flavored that I haven't liked. In fact, one of my favorite pumpkin-flavored things is O'Fallon Pumpkin Beer, which sadly was very limited last fall. I'm totally going to stock our fridge with it next fall.<br />
<br />
<br />
Almost done! Now it's on to step five, create eleven questions for the five blogs I nominated:<br />
<br />
<i>1. What is the one thing you love most about teaching?</i><br />
<i>2. What is the best gift you have ever received or given?</i><br />
<i>3. Outside of teaching, what is one thing you do to try and make the world a better place?</i><br />
<i>4. Who is your favorite U.S. president, and why?</i><br />
<i>5. What is your favorite "me time" activity?</i><br />
<i>6. What is your survival strategy for the imminent zombie apocalypse?</i><br />
<i>7. What "energizes" you during the school year?</i><br />
<i>8. What is your favorite holiday?</i><br />
<i>9. What one thing do you miss about your childhood?</i><br />
<i>10. Where do your lesson ideas come from?</i><br />
<i>11. What one thing do you want your students to remember you for?</i><br />
<br />
<br />
BAM! Pretty much done. All that's left now is to contact each of these bloggers and let them know of the torture that awaits them.<br />
<br />
This was a fun exercise! I certainly hope that I'll have more to blog about soon, but this is a good start.<br />
<br />
In the meantime, I am happy to say that I have accepted this nomination; and I can rest easy, knowing that I won't have to be looking over my shoulder for any blogging ninjas... for now.Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com3tag:blogger.com,1999:blog-3248548197809593245.post-86384366164543665442013-01-17T09:36:00.000-06:002013-01-17T11:52:46.548-06:00Congress Passes Gun Violence Law Banishing Children From Country Altogether<div style="font-family: Verdana,sans-serif;">
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<span style="font-size: x-large;"><b>CONGRESS PASSES GUN
VIOLENCE LAW BANISHING CHILDREN FROM COUNTRY ALTOGETHER</b></span><br />
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<span style="font-size: large;"><b>NRA strongly endorses
bipartisan legislation; President Obama to sign</b></span></div>
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<span style="font-size: small;">WASHINGTON, D.C. – In a stunning development on Capitol Hill
earlier today, Congress approved comprehensive legislation aimed at curtailing
gun violence against children by making it illegal for children to live in the
United States, period. </span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">President Barack Obama signaled that he would immediately sign the
bill, which passed with overwhelming bipartisan support in both the House and
the Senate. The National Rifle Association (NRA) also endorsed the bill.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">The law, titled the “Getting Them Far Overseas Act” (or the
GTFO Act), calls for the immediate deportation of children under the age of 18
to randomly assigned foreign countries.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">Families who are “conscientious objectors” to sending their
children to any foreign nation have the option to place them on a rickety old sailing
barge that will float aimlessly throughout international waters.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">Women who are currently pregnant will not need to worry
about unintentionally breaking the law when their new infants are born. A
section of the GTFO Act, popularly known as the “Nermal Clause,” stipulates
that all children born after January 15 will be mailed to Abu Dhabi within 72
hours of birth.</span></div>
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<br /></div>
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<span style="font-size: small;">In Washington, D.C., within minutes of the news that the
GTFO Act had passed, parents were already lining up their kids at Reagan
International Airport to send them off to their new homes.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“I saw daddy dancing in the hallway, hugging his hunting
rifle and saying ‘Thank God I don’t have to give <i>you</i> up,’” one 8-year-old boy said. “Then he grabbed a baseball
glove and took the rifle out to the backyard to play catch with it. Daddy never
played catch with <i>me</i>.”</span></div>
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<br /></div>
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<span style="font-size: small;">Reaction from the White House was considerably upbeat.</span></div>
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<br /></div>
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<span style="font-size: small;">“This is an historic day in the fight to end gun violence against
children in our country,” President Obama said, speaking to reporters from the
White House press conference room. “Working together with parties on all sides
of the issue, we were able to determine the best possible solution to this
problem; that solution is GTFO.” </span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“Obviously, there were a lot of folks around the country who
insisted on clinging to their guns,” the president noted. “But remarkably,
almost nobody was opposed to the idea of getting rid of our kids.”</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">President Obama tapped
Vice President Joe Biden last month to lead a task force addressing recent mass shootings in the United States. The task force spoke with
legislators, health care professionals, members of the NRA, officials from the
video game industry, and others about how to curtail gun violence against
children.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“At one point during our talks, I mentioned to everyone,
‘the common denomination here seems to be keeping our kids safe,’” Biden said,
speaking via conference call. “Then someone corrected me by saying, ‘Joe, I
think you mean common <i>denominator</i>.’ I
was never really that good with fractions.”</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">When the vice president realized that everyone involved in
the talks were concerned about the safety of the nation’s children, an
unorthodox proposal materialized.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“As we were finalizing our recommendations to the president,
we sort of had an epiphany,” Biden said. “We’ve tried banning guns, but there’s
a fat chance of that ever happening. So we asked ourselves, ‘what if we tried banning
kids instead?’ After all, there are two sides to gun violence: the guns, and
the victims. Eliminate one side and the problem goes away. We’ll
never get rid of guns, so we’re getting rid of the victims.”</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“This is a big [bleep]ing deal,” the vice president added.
“And I can say that freely, because there won’t be any impressionable young
children around to hear it. Hell, I can just say whatever the [bleep] I want
now. This is great!”</span></div>
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<br /></div>
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<span style="font-size: small;">House Speaker John Boehner had high praise for the
legislation.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“The true brilliance of this law is how it narrows the gap
in the deficit,” Boehner said. “With all of our children being sent overseas,
American taxpayers won’t be able to take any child credits on their returns, which
will generate billions in revenue. So not only do we solve the gun violence
problem, but we also eliminate a tax loophole that has long plagued our nation’s
fiscal health. Good riddance to those little brats, I say.” </span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">NRA president David Keene embraced the news that the GTFO
Act had passed.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“It was really the only sensible thing the government could
do,” Keene said. “I’m thrilled
that our government recognized the need to protect the 2<sup>nd</sup> Amendment
while at the same time keeping our children safe. It’s certainly not easy to
choose between your kids and your guns. I mean, the eleven SIG P229s I conceal
and carry at all times are just as much children to me as my <i>actual</i> children are. But we’re doing the
right thing.”</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">When asked how he thought America’s children would handle
the transition to living in other countries around the world, Keene showed
little concern.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“Naturally, we want to make sure our kids are safe as
they’re involuntarily scattered across the globe,” he said. “I would love to
let each of them take their own guns for protection. But, countries like Great
Britain and Australia have extremely tight restrictions on owning firearms, so
that’s not possible.” </span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“Fortunately, the rate of gun violence over in those countries
seems to be far lower than it is here,“ Keene added. “So our kids will be in a
much safer situation. Thank goodness they have their [bleep] together over
there.”</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">While international reaction to the GTFO Act appeared mixed,
U.N. Secretary-General Ban Ki-moon indicated that member nations would be ready
and willing to accept the incoming throngs of American children and help them adapt
to a life of exile.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“It’s not exactly the craziest thing the United States has
done, so whatever,” Ki-moon said.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">Meanwhile, President Obama contemplated how the law would affect
his own family.</span></div>
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<br /></div>
<div class="MsoNormal" style="font-family: Verdana,sans-serif;">
<span style="font-size: small;">“With Sasha and Malia gone, I won’t have to listen to those
goddamned ‘One Direction’ punks anymore,” the president said. “That’s what
makes this law beautiful.”</span></div>
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<span style="font-size: small;"><i>So yeah, today's entry is in the style of <a href="http://www.theonion.com/" target="_blank">The Onion</a>.</i> <i>If you found it even remotely humorous, I will consider this entry a tremendous success. Thanks for reading!</i></span></div>
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Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com11Chicago, IL, USA41.8781136 -87.62979819999998241.499532099999996 -88.275245199999986 42.2566951 -86.984351199999978tag:blogger.com,1999:blog-3248548197809593245.post-7591070585394094662013-01-07T14:44:00.002-06:002013-01-07T14:50:34.066-06:00Should I Even Bother Reviewing For Final Exams?Happy New Year, everyone!<br />
<br />
It's been a while since I last released one of my incoherent ramblings into the wild jungles of cyberspace, and I
have much to talk about, so expect to see a few more posts in the coming
days. (And if you don't see said posts materialize, please nag me until
I get them done.)<br />
<br />
(As a side note, as of this writing, my blog has about 25,000 views accumulated since my <a href="http://brennemath.blogspot.com/2012/07/hello-world.html" target="_blank">first post</a> in July. 20,000 of those views are attributed to a post I wrote in August about <a href="http://brennemath.blogspot.com/2012/08/ninjas-undeniably-awesome-but-student.html" target="_blank">the ninja board</a>. Apparently Google likes ninjas.) <br />
<br />
My school resumed classes today, and
1st semester final exams are coming up in a week and a half. That means
the time has come to start <i>reviewing for finals</i>.<br />
<br />
I've
been wondering about this lately, the idea of spending a week and a
half of class time reviewing for final exams. I'm not completely sure I
ever do it the right way. Actually, I often wonder if there even <i>is</i> a right way. <i> </i><br />
<br />
<i>Does it even do any good</i> to review for final exams?<br />
<br />
Every
semester, I take the last week and a half or so before final exams to
review with my students everything that we learned over the prior 16-17
weeks, tell them what kinds of questions to expect on the final exam and
how many, give them time to work on review packets/assignments/flaming
obstacle courses, etc. and so forth.<br />
<br />
I've tried various
ways of helping my students to take stock of what they learned (or were
supposed to learn) over the semester. We've done the "review for finals
process" as a project (with a rubric and everything) where students had
to develop and publish their own study guides. We've done the classic
"Jeopardy!"-style review game. We've done notecards that students were
allowed to use on the final. We've done review assignments with the
final exam questions <i>literally lifted from the exam itself</i>, with the numbers changed.<br />
<br />
And
what bothers me is this: Not once, that I can recall, in the four years
I've spent teaching so far, have I been able to discern whether or not
these methods of reviewing have done any good to any of my students.<br />
<br />
What <i>appears</i> to happen is that the students who more or less have been "getting it"
(or have been perpetually on the cusp of "getting it") all along are
best equipped to understand and solve the problems set before them on
the review assignments. Students who have been struggling all semester
-- for whatever reason -- also struggle to find success on review
assignments. It strikes me as a situation where the students who benefit
the most from reviewing for finals are also the ones who need it the
least, and the ones who benefit the least are the ones who need it the
most.<br />
<br />
I don't know why this appears to happen. (Or, if I'm really being honest, if it actually <i>is</i> what happens.)
Maybe I haven't been making enough of an effort to find out. Maybe it's some bizarre phenomenon that can't be explained, like Honey Boo Boo. Maybe I suck at teaching. (Okay, maybe not.)<br />
<br />
I was
discussing this matter with my lovely wife the other night, and she
asked me, "well, how do you know whether or not it's helping your
students?" I thought about it, couldn't come up with a great answer, got
childishly frustrated then stammered something like, "it's just based
on what I've observed in class, I don't know how to explain it!" Then I
pouted and decided to go do something else, because I'm <i>so mature</i>.<br />
<br />
The bottom line is, I've
never really been confident in my approach to reviewing for finals. I haven't made it easy for myself to tell whether or not my approach has a positive (or negative) effect. Maybe that's what makes me wonder if reviewing does any actual good.<br />
<br />
Perhaps in the
naivety of being a young teacher, I've been thinking of it the wrong
way. I think the best way to describe how I've approached reviewing for
finals is that I've seen it as an eleventh-hour scaffolding activity,
intended to give students one last hope at having a mathematical epiphany, a lifeboat that will float them safely through the perilous, shark-infested tides of the final exam.<br />
<br />
It never seems to really work that way. No lifeboats. Sharks with happy tummies.<br />
<br />
Maybe I <i>should</i> be looking at reviewing for the final exam as part of the cumulative assessment itself. Reviewing should really be more of a time for reflection and fine-tuning, not making a last-ditch effort for comprehending something for the first time. That's not to say there won't be a few students that <i>do</i> get that benefit from reviewing, but that shouldn't be the point. The point should be to look back at all of the work we've done all semester, take stock of what we've learned and what we still have questions about, address areas that still need addressed, and perhaps even celebrate.<br />
<br />
My angst aside, here's what I'm trying this time around. The other day, I remembered something I read on <a href="http://deltascape.blogspot.com/2012/10/what-if-we-gave-the-answers.html" target="_blank">David Coffey's blog</a> about <a href="http://deltascape.blogspot.com/2012/10/what-if-we-gave-the-answers.html" target="_blank">giving students the answers</a> to the problems and having them explain <i>how</i> to get that answer. In my case, I'm going to provide students with a set of problems that are similar to what's on the final exam, give them all of the answers, and require them to explain how to get each answer. This way, they focus on <i>how to solve the problem</i> as opposed to focusing on <i>getting the right answer</i>. <br />
<br />
I don't really know if this will be any better or any worse than what I've tried in the past. But, I think it will at least alleviate some of the anxiety and second-guessing that comes with reviewing for final exams. We have a week and a half, which should be plenty of time to address any questions or concerns that arise as students work through their review assignments, particularly since I am putting the focus on articulating their mathematical thought processes.<br />
<br />
Will it do any good? Your guess is as good as mine.<br />
<br />
<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com9tag:blogger.com,1999:blog-3248548197809593245.post-23763263119821200722012-09-14T14:23:00.004-05:002012-09-14T14:24:54.533-05:00Sometimes It's Good to Take a DetourProbably one of the coolest things about teaching is when a student asks a really good question that lets you detour from your original plan to talk about something really super-awesome.<br />
<br />
That happened in my class today.<br />
<br />
We were discussing slope and going through a few example problems with the slope formula. I decided to show them one example that resulted in an undefined slope. I gave them the points (7, 3) and (7, 10), then we worked through the problem. We got to a point where we had 7/0 on the board and I asked the students what that meant. The consensus was that the slope was undefined because "we can't divide by zero."<br />
<br />
Then, one of my students asked: "Mr. Brenneman, why can't we divide by zero?"<br />
<br />
I stopped. I looked at him. I said, "I <i>love</i> that question! Let's put aside what we're doing and talk about <i>this!</i>"<br />
<br />
I then launched into a brief explanation of proof by contradiction and asked them to put aside the laws of mathematics for one second. "Let's suppose that you <i>can</i> divide by zero," I said. "Let's consider what 0/0 would be equal to. What do you think?"<br />
<br />
Many students chimed in with "0." Others chimed in with "1." I asked each side to back up their reasoning.<br />
<br />
"Well, it would be zero because you're dividing zero by another number," one student said.<br />
<br />
"I think it would be one, because 2/2 is 1, 4/4 is 1, so 0/0 would be 1," said another.<br />
<br />
A few minds were blown when I told them they were <i>both</i> right.<br />
<br />
Here's why:<br />
<br />
Assuming we can divide by zero, the quotient of 0/0 yields two distinct yet equally valid results.<br />
<br />
Suppose we choose a number <i>a</i> from all of the numbers in existence. We say that 0/<i>a</i> = 0 (the <b>zero property </b>of division) and <i>a</i>/<i>a</i> = 1 (a form of the <b>multiplicative inverse property</b>).<br />
<br />
In this scenario, division by zero is allowable. (This is an important distinction, because normally the two properties I mentioned above specify that <i>a</i> must be nonzero.) So, 0/0 = 0 by the zero property. But, 0/0 = 1 by the multiplicative inverse property.<br />
<br />
Thus, it is reasonable to conclude that 0/0 = 0 <i>and</i> 0/0 = 1.<br />
<br />
In other words, 0 = 1.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjE3rigJjWKis9fbab2-qeKwbSr04EZDn0tP4OvD48kdn-BAtdrm9MUnq3nCWikVJrQcSvIajMmYUraIlaquz8FTRQnBTOOHe26Gk9RgCA3j2y28xSeQoJsOy6LwgeB_eX1g8qxHsr3BwZ0/s1600/2012-09-14+08.22.18.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjE3rigJjWKis9fbab2-qeKwbSr04EZDn0tP4OvD48kdn-BAtdrm9MUnq3nCWikVJrQcSvIajMmYUraIlaquz8FTRQnBTOOHe26Gk9RgCA3j2y28xSeQoJsOy6LwgeB_eX1g8qxHsr3BwZ0/s640/2012-09-14+08.22.18.jpg" width="480" /></a></div>
<br />
The discussion can certainly stop here, because we have arrived at a conclusion that is mathematically absurd. Furthermore, this absurdity stems from the initial assumption that we <i>can</i> divide by zero; hence, we must conclude that we <i>cannot</i> divide by zero.<br />
<br />
But I knew that ending our discussion at 0 = 1 wouldn't have been nearly quite as fun as proceeding with even more absurdity.<br />
<br />
So, I asked the students, "what would 1 + 1 be equal to?"<br />
<br />
Many said 2. Some said 1. They were all correct. I showed them why.<br />
<br />
1 + 1 certainly equals 2. But, we've already established that 1 = 0, so we can also say that 1 + 1 = 1 + 0 = 1. Or, 1 + 1 = 0 + 0 = 0.<br />
<br />
In other words, 0 = 1 = 2.<br />
<br />
I extended it one more time by asking the students what 1 + 1 + 1 would equal. Some said 3, some said 2, some said 1. Again, they were all correct. Using similar reasoning as the "1 + 1" case, we concluded that 0 = 1 = 2 = 3.<br />
<br />
At that point, the students came to realize that if we kept going, eventually we would conclude that <i>all numbers would be equal to each other</i>.<br />
<br />
I told the students one of my favorite mathematically absurd things to say: "If Congress legalized division by zero, we could solve all of our economic problems. We wouldn't have a $15 trillion debt, because if we can divide by zero then 15 trillion would be equal to zero. We wouldn't owe anyone $15 trillion. Problem solved!"<br />
<br />
My students seemed to love it. Sometimes it's fun to drop what we're doing and discuss something far more interesting when the opportunity arises.<br />
<br />
<br />
<br />
<br />Anonymoushttp://www.blogger.com/profile/09025084422261929614noreply@blogger.com2