tag:blogger.com,1999:blog-57792713852566255332018-05-21T05:31:49.437-07:00cheesemonkey wondersrenegotiating the didactic contractcheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.comBlogger180125tag:blogger.com,1999:blog-5779271385256625533.post-49930566226393534882018-04-06T12:21:00.002-07:002018-04-06T12:21:13.296-07:00HERESY WARNING: Breaking through on quadratic function analysis & graph sketchingOver the last two days, I've had a few important breakthroughs with discouraged Algebra 1 learners about quadratic functions and their graphs. I wanted to document this for myself before I have a chance to forget about it for next year.<br /><br />If you have rigid beliefs about the only ways for students to approach quadratic functions, analysis, and graphs, then this post is definitely not for you.<br /><br />Consider yourselves warned.<br /><br />My most discouraged Algebra 1 learners are extremely gifted kids, but this year's crop are definitely dreamers, not stalkers. They need to really marinate in something for a long time before it takes root in their minds. They are factoring warriors, but the quadratic formula and complicated answers can be really daunting for them.<br /><br /><a href="https://2.bp.blogspot.com/-BRGVWacomxw/WsfGvX8ZbUI/AAAAAAAABAk/ri-D3zzjitgZRpNoCjDsZkjdkY1g7M88wCLcBGAs/s1600/graphing%2Bquadratics.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-BRGVWacomxw/WsfGvX8ZbUI/AAAAAAAABAk/ri-D3zzjitgZRpNoCjDsZkjdkY1g7M88wCLcBGAs/s320/graphing%2Bquadratics.jpg" width="240" /></a>The connection I have wanted all of my students to make between quadratic functions and their graphs is this: even when a quadratic doesn't have neat, simple, integer answers, it still has a number of neat, simple aspects that they can grab hold of. There will always be an axis of symmetry. There will always be a vertex.<br /><br />What I have discovered — or rather, what they have been teaching me all this week — is that if you approach a quadratic function from the understanding that you already <i style="font-weight: bold;">know</i> how to find the AOS and the vertex, you can make a TON of important discoveries and understandings about its graph and the many properties of the function that are important to understand.<br /><br />What they taught me today is that they know how to use the axis of symmetry formula like a key in a lock. They can identify <i>a</i>, <i>b</i>, and <i>c</i> in a quadratic, and they understand that they can find and graph the AOS equation quickly and fearlessly. Then they can plug in the value they found for the axis of symmetry to identify the vertex of the parabola.<br /><br />From there, it is a simple matter to find some more points for your sketch and to identify their mirror reflections across the axis of symmetry.<br /><br />The beauty of their method is that it makes it easy for them to develop a meaningful conjecture about whether or not a quadratic function has any zeros.<br /><br />If the parabola is floating above the x-axis, then they can use the QF to confirm their hunch that there are no real-number zeros to the function. Likewise if the parabola is submerged below the x-axis.<br /><br />I love that they naturally figured out today what it means for the vertex to be a maximum or a minimum.<br /><br />And I especially love the fact that they made these discoveries themselves.<br /><br />They still don't understand how to complete the square or use the quadratic formula to blast through problem after problem to find complicated zeros of non-trivial quadratics.<br /><br />But this feels less important to me than the fact that they have made important connections and developed their own methods for investigating quadratic functions. And it has been an important reminder to me to design learning experiences that empower them to make these connections and discoveries for themselves.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com3tag:blogger.com,1999:blog-5779271385256625533.post-30167325498920827592018-01-28T09:55:00.002-08:002018-01-28T09:56:49.219-08:00The Centrality of the Cross-Racial Therapeutic Alliance in My Classroom<br />As part of our school's five-year program of equity transformation, we the staff engage in monthly two-hour seminars focused on issues of equity and diversity. As someone who is leading (and learning to lead) equity transformation in my very large school, I live on both sides of the fence in this process. As a peer, I am often a witness to frustration and discouragement that naturally arise in an inherently vexed and vexing process, since we are only in year two of the process. And as someone on the leadership team, I am often tasked with encouraging greater engagement in a program whose theory of action is I sometimes have issues of practice with.<br /><div><br /></div><div>Nevertheless, I wanted to explore my own understanding of an important part of this month's reading included an article titled, "Racial Microaggressions in Everyday Life: Implications for Clinical Practice," by Derald Wing Sue et al (May—June 2007, <i>American Psychologist</i>). </div><div><br /></div><div>Most of my colleagues' responses to this piece have centered on the numbingly detailed taxonomies the authors present about forms of microaggressions, microassaults, and microinvalidations. This part of the article felt deeply Aristotelian to me — busily categorizing, classifying, analyzing, exemplifying, and naming these forms of everyday racist aggressions that occur every day everywhere in our country.</div><div><br /></div><div>What has struck me most in this article — and what has stayed with me — was the discussion toward the end of the piece about the <b><i>centrality of the therapeutic alliance within cross-racial dyads</i></b>. This hit home for me because it is what I experience as the most central part of my own work in my classroom with my students of color, so I noticed myself sitting up straighter and taking much better notes when I was reading this section.</div><div><br /></div><div>A lot of research on the effectiveness (or ineffectiveness) of the therapeutic process has zeroed in on the <b><i>therapeutic alliance</i></b> as the most essential determinant of success in personal transformation. In therapy, this means that the client <i style="font-weight: bold;">must</i> perceive the therapist himself or herself as being both deeply trustworthy and expert in the process of transformation. It seems to me that this same relationship exists within the classroom. The bond I have to form with my students requires that they perceive me as both worthy of their trust and expert in the process of transformation. </div><div><br /></div><div>In addition to the client's perception of the therapist, the client must also have a very specific perception of him- or herself within the therapeutic context — namely, they must perceive that they themselves are being deeply understood and received in a relationship of unconditional positive regard. This too is a critical part of my own culturally responsive pedagogy. My students must perceive themselves as being welcomed to "come as you are" in my classroom because I know that this is the only thing that will make them feel safe enough to become vulnerable enough to engage in the transformative mathematical identity work that we do in my classes.</div><div><br /></div><div>A huge part of my own practice of trustworthiness with my students is about being open to accepting responsibility for harms that I cause (or perpetuate) and for taking corrective actions to repair the harms and clean up my side of the street in our relationships. This becomes even more important as a prerequisite for cultural competence, because I am sure there are a zillion unconscious ways big and small in which I, as a White teacher microassault, microinsult, or microinvalidate my students of color while at the same time using every tool in my power to love and support them. Being in community is like being in any relationship. You can never fully <b><i>avoid</i></b> running into conflict, but you <i style="font-weight: bold;">can</i> learn better ways to repair the local harms that arise so that the underlying fabric of the relationship remains intact for the benefit of all parties.</div><div><br /></div><div>Because most helping professionals, including teachers, experience themselves as fair and decent people who are incapable of causing deliberate racist harm toward our students of color, it can be difficult to see microaggressions as they are happening because they are so often invisible to authority figures who are accustomed to swimming in a soup of white privilege. Yet I must continue to practice noticing my own white racial identity in order to develop this capacity in my conscious mind and social practice.</div><div><br /></div><div>For me, the most urgent idea in this article was the fact that when racial microaggressions continue to occur within a cross-racial therapeutic context without being noticed, named, and addressed, clients of color are at risk of not continuing in their own work of inner development. To put it in the context of my classroom, if students perceive that I perceive myself to have no responsibility for noticing, naming, and addressing racial injustices at the micro- level within our relationship, then they are more likely to become discouraged and give up.</div><div><br /></div><div>I don't know where all this thinking is going to take me. I only know that framing the cross-racial teaching and learning relationship as a therapeutic alliance feels transformative to me. And that means I need to keep investigating my role in it.</div>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com1tag:blogger.com,1999:blog-5779271385256625533.post-62715330240483086022017-09-17T10:22:00.002-07:002017-09-17T10:23:41.113-07:00Conversation With Grace: Why Strive For Diversity?Ever since I read <a href="http://blog.graceachen.com/2017/08/why-strive-for-diversity.html">Grace's post</a>, I've been struggling.<br /><br />I have nothing but bad, inadequate answers to this question. And since its premise lies at the heart of my every day's work, I have felt stuck as I struggle to find a better answer.<br /><br />I still have only terrible answers, but I need to get unstuck so I can get going again. So I'm writing this post as a chance to stick a pin in the best bad answers I have right now.<br /><br />1. <b><i>tikkun olam</i></b><br /><i>Tikkun olam</i> is the central assignment in the spiritual curriculum I grew up in. It is the requirement that we work actively to heal the world. When I was a child in the 60s, <a href="https://en.wikipedia.org/wiki/Israel_S._Dresner">our rabbi</a> was an active figure in the civil rights movement — a friend and colleague of Dr. King's, one of the Freedom Riders, and "the most arrested rabbi in America." As Dogen Zenji, the founder of the Zen lineage said, "When you walk in the mist, you get wet." Spending day after day, week after week in this community, I walked in the mist. I got wet. I came to believe in the power of <i>tikkun olam</i> to make America a more just place, even if it was still massively imperfect. <i>Tikkun olam</i> also taught me that if we as a society have the <i>possibility</i> of providing quality education and access to power for some, then we need to make sure that we make that <i>possibility</i> a <i>reality</i>. Our society and our world are a mess right now, so there's no time to lose. And the only way to heal the world is to actively work toward healing the world. I have to accept that I may be doing it wrong or badly, but in making positive effort for the good each day, I am doing the best that I can. And the more people I reach, the better. Because I never know who will trigger...<br /><br />2. <b><i>the hundredth monkey principle</i></b><br />In the 1960s and 70s, the hundredth monkey principle became an archetypal myth and story for me about how critical mass seems to come about in human society. The story has been used and recycled in many different contexts, but what strikes me as important is this: the story provides a useful metaphor for me about what happens as we share good ideas and carry them with us. For me, the big idea is that we can have no idea where or with whom the tipping point or critical mass will happen. So sharing widely — and diversely — is important to me because I have no idea who will be that "hundredth monkey." Identifying the hundredth monkey is above my pay grade. My job is to share best practices and wisdom so that it <i style="font-weight: bold;">reaches</i> the hundredth monkey and we can achieve some degree of <i>tikkun olam</i> sooner rather than later. The more people I reach, the better.<br /><br />3. <b><i>diversifying the power structure</i></b><br />I was grateful that Julie Wright shared <a href="https://www.nbcnews.com/video/michelle-obama-shares-her-takeaway-from-hidden-figures-834318403859">this clip of Michelle Obama</a> sharing her beliefs about the power of diversifying the power structure after watching the film <i>Hidden Figures</i>. Michelle and I were at Princeton at the same time, and we both benefited enormously from its diversity initiatives. I was in only the tenth class of women at Princeton, and Michelle was two years behind me. The education we received there was transformative, but it was also transformative to live in community with others from such dramatically differing backgrounds from around the world. It was my first experience of gifted education, and it changed me. It changed a lot of us. Several years later, I was privileged to be a part of a conversation at a conference in which Henry Louis Gates, Jr. framed the need to diversify the power structure and the power elite if we want true transformation, and I believe that for better or for worse, he is right. If we want to transform our society, we need to transform its power structure, and for that, we need to diversify our classrooms. As I've gotten older, I've gotten less apologetic for my elitist tactics here. I have less time to waste, and I want a better quality of leaders across the board in our pipelines.<br /><br />4. <b><i>Yes, it is better for me.</i></b><br />This is the argument I feel most ashamed of, but I have to be ruthlessly honest about my own motives. I hate it when I have to admit that I am ruled by enlightened self-interest, but it's a reality in economics and it's often a reality of my own life. It has nothing to do with pity for others and everything to do with what makes my own experience in the world I live in better. I love exploring the world around me, and that means I love experiencing viewpoints and perspectives that are different from my own. In my previous career in high tech, I experienced how blind spots take over when everybody at the table looks the same. In education, I see how the wealthy and powerful push idiotic agendas when they only listen to other wealthy and powerful people.<br /><br />So yes, I work for greater anti-racist diversity in my educational context because it is better and healthier and saner for me.<br /><br />I have a strong suspicion it is also better for others, but by the principles of <i>tikkun olam</i>, I am restricted to cleaning things up on my own side of the street.<br /><br />I would love for us to have ever more diverse TMCs, but I am also bound to respect the differing boundaries, needs, and wants of the people who would make those things more "diverse" for people like me. I am bound to respect that for those who spend their school year isolated within an oppressively dominant culture, it may <i style="font-weight: bold;">not</i> feel optimal <i style="font-weight: bold;">to them</i> to spend their summer resources attending a PD conference that is still largely populated by teachers from the dominant cultures. That puts the onus on me and others like me to strive to make TMC a worthwhile and valuable choice for them to make. But it's critical to recognize and respect that those who would make a TMC "more diverse" place for <i>me</i> have <i><b>their <u>own</u> needs, wants, and priorities</b></i>. I need to make sure I am not trying to manipulate others into serving my own needs, wants and priorities.<br /><br />I cannot bear to #<i>pushsend</i>, but I also can't bear to feel stuck any more.<br /><br />I want to get back to giving my wild and imperfectly diverse classes the best educational opportunities I can give them, so I'm going to #<i>pushsend</i> anyway. God help me.<br /><br />cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-55707144811825006562017-08-24T16:16:00.001-07:002017-08-24T16:16:37.844-07:00Why does this feel so dangerous...?Maybe it's just the cold medicine speaking (I am battling a wicked cold), but today in 6th block Algebra 1, as we were working on Exeter problems, I ditched my usual, "safe" piece of lecture and went went for the jugular.<br /><br />I wanted to give them the real stuff — not the party line position.<br /><br />I advocated for radical, student-centered sense-making.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-EV7WsG-TesY/WZ9dTvOxnpI/AAAAAAAAA_8/9RKMCE7eTQothmF4eV3BE1Dxt4cNxBM3gCLcBGAs/s1600/IMG_4918.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://3.bp.blogspot.com/-EV7WsG-TesY/WZ9dTvOxnpI/AAAAAAAAA_8/9RKMCE7eTQothmF4eV3BE1Dxt4cNxBM3gCLcBGAs/s320/IMG_4918.JPG" width="240" /></a><a href="https://2.bp.blogspot.com/-hOKIqIrELDY/WZ9dIYzE3xI/AAAAAAAAA_4/udE4UAds2YEl0MDH7AOkliCDYDFHFSfNgCEwYBhgL/s1600/IMG_4919.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://2.bp.blogspot.com/-hOKIqIrELDY/WZ9dIYzE3xI/AAAAAAAAA_4/udE4UAds2YEl0MDH7AOkliCDYDFHFSfNgCEwYBhgL/s320/IMG_4919.JPG" width="240" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">WHY DOES THIS FEEL SO DANGEROUS?????</div><span id="goog_1266781731"></span><span id="goog_1266781732"></span><br />cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-62317093541698805402017-07-22T13:23:00.003-07:002017-07-22T13:25:54.053-07:00"You are what you seek," the wise one said.Once upon a time, some time late in 2011, there were some lonely, kooky, determined math teachers trying to get better.<br /><br />They searched blogs and joined Twitter in their quests, and eventually they found some like-minded spirits on the internets who were also questing.<br /><br />In 2012, forty of us decided to meet in person in St. Louis and hold our own conference. Nobody outside that first group (besides Fawn, who hosted #TwitterJealousyCamp) knew or much cared about what we were doing. We were doing it because we wanted to do it. Period.<br /><br /><i>FUN FACT: Out of forty attendees at the first TMC, I was the one and only attendee from California. In fact, we had more attendees from Mississippi than from California</i>.<br /><br />It wasn't perfect, but it was real — and that kindled a spark. What made it magical was the fact that people showed up and brought their A game. I learned something amazing from every single person at that conference.<br /><br />So if you attending TMC for the first time this year, please temper your freaking out with the knowledge that we started this thing because we were looking for YOU. We are STILL looking for you.<br /><br />Before TMC, I always think of one of my favorite quotes from the great Jungian psychoanalyst and storyteller Clarissa Pinkola Estes:<br /><blockquote class="tr_bq"><i> Even though there are negative aspects to it, the wild psyche can endure exile. It makes us yearn that much more to free our own true nature and causes us to long for a culture that goes with it. Even this yearning, this longing makes a person go on. It makes a [person] go on looking, and if she cannot find the culture that encourages her, then she usually decides to construct it herself. And that is good, for if she builds it, others who have been looking for a long time will mysteriously arrive one day enthusiastically proclaiming that they have been looking for this all along.</i></blockquote>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-39899879370054171082017-07-21T14:08:00.003-07:002017-07-21T14:08:52.063-07:00#TMC17 MORNING SESSION OVERVIEW: Differentiating CCSS Algebra 1 — from drab to fab using Exeter Math 1 & Exploratory Talk<br /><div>How can we use a problem-based throughline such as Exeter to transform a lifeless but mandated Algebra 1 course into a rich, differentiated experience of mathematical sense-making for a wide range of students?</div><div><br /></div><div>This morning session will be a master class in differentiating a generic Algebra 1 course using problem-based learning, exploratory talk, PCMI-style differentiation, and deliberate practice in its appropriate place together with metacognitive self-monitoring.</div><div><br /></div><div><div>Over the course of our time together, we will move back and forth between the perspective of learners and the perspective of teachers. During reflective<b style="font-style: italic;"> "master class" segments</b>, we will explore the theories, techniques, and practical aspects of rearchitecting Algebra 1. During immersive <b><i>“math-doing</i></b>” <b><i>segments</i></b>, we will do selected sequences of problem-based mathematics together in groups so we can experience different approaches to concept development, cultivating habits of mind, building norms through math content, and engaging the whole student through experiential problems. Immersive segments will be interwoven with reflective, “master class” segments in which we will analyze the theories, techniques, and ideas we're exploring.</div></div><div><br /></div><div>Here is the 30,000-foot overview of the topics we'll be digging into over our three days.</div><div><br /></div><div><b><span style="font-size: x-small;"><u>DAY 1 TOPICS</u></span></b></div><div><b><span style="font-size: x-small;"><br /></span></b></div><div><b><span style="font-size: x-small;">I. THEORETICAL FRAMEWORK:</span></b></div><blockquote class="tr_bq">Brief review of the How People Learn (HPL) learning cycle so that we will have a shared vocabulary for our work together. <i>EQ: What research research informs these ideas about teaching and learning with understanding?</i></blockquote><div><b><span style="font-size: x-small;">II. TIPS ON PRACTICAL PREPARATION FOR TEACHING EXETER:</span></b></div><blockquote class="tr_bq">Practical strategies and tips for organizing and managing your own teaching and learning of Exeter sequences to support your work with students. <i>EQ: This feels overwhelming— how can I set myself up for success?</i></blockquote><div><b><span style="font-size: x-small;">III. REIMAGINING ALGEBRA 1 AS A COURSE IN ADVANCED PROPORTIONAL REASONING:</span></b></div><blockquote class="tr_bq">Why and how Algebra 1 must be reimagined as a course in advanced proportional reasoning. </blockquote><div><b><span style="font-size: x-small;">IV. EXETER'S BEST-KEPT SECRET—EXPERIENTIAL, NOT EXPERIMENTAL:</span></b></div><blockquote class="tr_bq">Through experiential "doing" segments and reflective discussions, we'll explore some of the ways in which the anchor problems and supporting problems within the Exeter sequences encourage students to get inside the problems in a state of flow rather than killing time filling in charts with mindless data-gathering. <i>EQ: How do the Exeter problems cultivate a stance of shareable curiosity?</i></blockquote><div><b><span style="font-size: x-small;">----------------------------------------------------</span></b></div><div><b><span style="font-size: x-small;">DAY 2 TOPICS</span></b></div><div><b><span style="font-size: x-small;"><br /></span></b></div><div><b><span style="font-size: x-small;">V. EXPLORATORY TALK AS THE GROUND:</span></b></div><blockquote class="tr_bq">Strategies for integrating Talking Points as a focused technique for developing collaborative speaking and listening skills. </blockquote> <b><span style="font-size: x-small;">VI. RADICAL DIFFERENTIATION—THE BOWEN & DARRYL METHOD:</span></b><br /><blockquote class="tr_bq">Structuring your room and tasks to support the needs of both katamari and speed demons. <i>EQ: How can I create an environment that consistently values all student ideas and thinking?</i></blockquote><div><div><b><span style="font-size: x-small;">----------------------------------------------------</span></b></div><div><b><span style="font-size: x-small;">DAY 3 TOPICS</span></b></div><div><b><span style="font-size: x-small;"><br /></span></b></div><div><b><span style="font-size: x-small;">VII. ANCHORING MATHEMATICS IN THE PRESENT MOMENT:</span></b></div><blockquote class="tr_bq">Anchoring students' mathematical thinking in the body, in the present moment, and in the value of their own existing knowledge and understanding.</blockquote><div><b><span style="font-size: x-small;">VIII. A PLACE FOR PRACTICE ACTIVITIES:</span></b></div><blockquote class="tr_bq">How and where to integrate practice activities in ways that support student agency, dignity, and understanding.</blockquote></div>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-18072580760199420732017-07-07T09:36:00.000-07:002017-07-07T13:35:41.011-07:00Things That Work #1: Regular Vocab Quizzes in GeometryOne of the things that worked incredibly well last year—and which I want to extend this year—is regular vocab quizzes in Geometry.<br /><br />Vocabulary is the gating factor for success in a problem-based, student-centered Geometry class. If you can't <i style="font-weight: bold;">talk</i> about geometry, you can't <i style="font-weight: bold;">collaborate</i> about geometry.<br /><br />I learned the value of extremely routine-looking vocabulary quizzes when I taught 8th grade English with Alec MacKenzie, Linda Grady, and Kelly Starnes. At the beginning of the school year, the copy room delivered us each a giant stack of very basic matching quizzes: numbered terms in the left-hand column, lettered definitions on the right. Each student got a vocabulary workbook at the beginning of the school year. Every week we assigned a new chapter/list. Every week we gave a matching quiz. And then we would trade and grade them.<br /><br />At some level, I recognize that this sounds stultifying. But at another level, it was incredibly empowering for the students. Everybody understood <i style="font-weight: bold;">exactly</i> what was being asked and expected. And everybody saw it as an opportunity to earn free points. Students gave each other encouraging written comments and cheered each other on. They saw their scores as information—not as judgment. They used what they knew to make flash cards or Quizlet stacks. They quizzed each other. They helped each other.<br /><br />And nobody ever complained about the regularly scheduled vocab quiz. It was a ritual of our course.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-gMtYWvxlGm0/WV-0wDQ8goI/AAAAAAAAA_Q/P-kn_eEhG1YU0ery5eyK-iicex4ztzRNgCEwYBhgL/s1600/Chapter%2B7%2BVocabulary%2BQuiz.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1237" height="320" src="https://1.bp.blogspot.com/-gMtYWvxlGm0/WV-0wDQ8goI/AAAAAAAAA_Q/P-kn_eEhG1YU0ery5eyK-iicex4ztzRNgCEwYBhgL/s320/Chapter%2B7%2BVocabulary%2BQuiz.png" width="247" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Vocab quiz for initial unit on circles</i></td></tr></tbody></table>In my first few years of teaching Geometry, I have noticed that the kids who make the effort to integrate and use the vocabulary and specialized terms tend to succeed. And the kids who don't use the language of geometry suffer. So I decided to use what I know to raise the number of kids who know and use the vocabulary by instituting regular vocabulary quizzes for the relevant lessons or chapters as we go.<br /><br />Many of my discouraged math learners sprang to life when I assigned this task. They pulled out flash cards, folded sheets of binder paper in half lengthwise, and started organizing the information they wanted to integrate. In most of my classes, I noticed that the highest-status math students often seemed to get stuck while the weaker students knew EXACTLY where to start and what to do.<br /><br />It was a revelation.<br /><br />It also ensured that everybody spent a little quality time on the focus task of preparing for the vocab quiz on Thursday or Friday. And this, in turn, meant that everybody was a little more ready to use the correct and appropriate mathematical vocabulary in our work. They noticed more because the <i style="font-weight: bold;">owned</i> more.<br /><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">Because these were "for a grade," kids put their shoulder into it. My colleagues in other departments commented about my students taking two or three available minutes during passing period to quiz each other. It gave them hope.</div><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">Now I want to create a full set of vocab quizzes for my whole year. </div><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">A few implementation notes:</div><div class="" style="clear: both; text-align: left;"></div><ul><li>I collect and shred/recycle all of the quizzes after I enter their scores so I can reuse the same quizzes from year to year. If I don't have your quiz, you can't get a score. I am strict about this.</li><li>Every new vocabulary term does not have to get quizzed, but lessons or units where there is a huge vocabulary burden that gets front-loaded deserves its own vocab quiz. I have been surprised to discover how many lessons are more vocabulary-intensive/language-intensive than I had realized.</li><li>Correct use of technical language is self-reinforcing. Once I introduce a new term, I mercilessly ask kids to remind each other of the definitions for 15 seconds in their table groups. Getting <i style="font-weight: bold;">one</i> kid to call out the correct definition to the whole class is not the point here. Getting 36 kids to all speak the definitions or the terms in their table groups is.</li></ul><b><span style="font-size: large;">UPDATE</span></b>: <b><i>D'OH! </i></b>I can't believe I forgot the most important implementation note I wanted to remind myself about!!!<br /><div><ul><li>There should be <b><i>many more</i></b> definitions in your right-hand list than there are terms in your left-hand list. Also definitions can be re-used. This way there isn't a zero-sum outcome if someone misses an answer.</li></ul><br /><div class="separator" style="clear: both; text-align: center;"></div></div>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com11tag:blogger.com,1999:blog-5779271385256625533.post-34985435984600800242017-05-17T09:46:00.001-07:002017-05-17T09:48:17.287-07:00Take Time to Save Time – Hall of Fame reference sheetsInevitably, teachers get known for their mottos. <a href="https://samjshah.com/2011/02/03/calculus-mottos/">Sam's mottos</a> are justifiably world-famous. Personally, I love "Don't be a hero." Mine are known mostly around my school, but it is interesting to see how they trickle down into students' unconscious minds.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-MaCIemZyhAc/WRx82QbikeI/AAAAAAAAA-E/L2iXLJUZReEN5M1HIemMka2ZEIDJrgjCACLcB/s1600/color%2Btells%2Bthe%2Bstory%2Bimage.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="307" src="https://3.bp.blogspot.com/-MaCIemZyhAc/WRx82QbikeI/AAAAAAAAA-E/L2iXLJUZReEN5M1HIemMka2ZEIDJrgjCACLcB/s320/color%2Btells%2Bthe%2Bstory%2Bimage.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Color telling the story</i></td></tr></tbody></table>Mottos pay off. My favorite is one I stole from my former colleague Alex Wilson: "Color tells the story." I don't understand how anybody can do math at a deep conceptual level without colored pencils. Color really does tell the story, especially in Geometry (see popular worked example at right).<br /><br /><br />One of my best math class mottos comes from published patterns for knitting. It is, "Take time to save time." In knitting, this means to make sure that the tension of your actual knitted work — your hands, your needles, your yarn — match the tension or gauge described in the knitting pattern. There are no shortcuts here. My knitting gauge tends to be extremely big or loose compared to most pattern-makers. I often have to use much smaller needles than specified in order to achieve a good match with the specified knitting gauge.<br /><br />In my classroom, "Take time to save time" means, synthesize your learning into a reference sheet. For all tests but the final, I allow students to have and make a half-page reference sheet. The first rule is, you can have anything you want except a photocopy of my work on your reference half-sheet. The second rule is, if you have more than a half sheet of 8.5 x 11 inch paper, then I get to tear it in half and choose which half you get. This rule gets tested even when I emphasize it. Every year somebody tests this rule. "But Dr. S! I only wrote a half-page worth of stuff on the paper!" It doesn't matter. I usually rip the whole thing lengthwise so they only get the right-hand half of the paper.<br /><br />It makes its point.<br /><br />In knitting, this point gets made by the scale and size of your finished object. If you insist on not checking your gauge, at some point, you will end up with a finger-puppet-sized sweater or a scarf the size of Lake Tahoe.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-xB_MV_eTh5I/WRx9_2qeGoI/AAAAAAAAA-M/R9_gM53-BS0LFkq7lF1aCJLOvLlm6q0pwCLcB/s1600/hall%2Bof%2Bfame%2Breference%2Bsheet%2Bexample.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://2.bp.blogspot.com/-xB_MV_eTh5I/WRx9_2qeGoI/AAAAAAAAA-M/R9_gM53-BS0LFkq7lF1aCJLOvLlm6q0pwCLcB/s320/hall%2Bof%2Bfame%2Breference%2Bsheet%2Bexample.png" width="186" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i>Clearly this student is going to ace the final.</i></td></tr></tbody></table>In our classes, this point gets made by your performance on our common final exam. Students who have been practicing making clear, concise, summaries and examples of their work and key points tend to turn in consistently strong performances. So on the final, I allow a full-page reference sheet (both sides). I emphatically <u style="font-style: italic; font-weight: bold;">want</u> students to consolidate their understanding and create their own examples. <b><i><u>That</u></i></b> is where the learning happens.<br /><br />So I was thrilled today when I asked to see examples of in-progress reference sheets. Many of them made my Hall Of Fame request to scan for posterity. This Algebra 1 student has totally nailed her understanding of mixture problems. This is the best example I've seen of a student consolidating her understanding of these modeling challenges.<br /><br /><br />cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-68148140430099320712016-12-01T09:07:00.001-08:002016-12-01T09:11:44.707-08:00The Festival of Reassessment (SBG)December is when I am truly grateful for strong routines. They mean I can split the class up and still count on everything moving forward. I am running behind in my pacing, so I needed to set up a mass SBG reassessment opportunity for slope skills yesterday. I set up all the reassessors along the window side of the classroom and all of the non-reassessors on the hallway side of the room. The non-reassessors worked on linear systems skills and problem-based learning while the reassessors worked on demonstrating mastery of slope skills.<br /><br />Whenever a reassessing student finished their work, they brought it up to be rechecked. I went through it right then and there while they watched. “Uh huh... uh huh... uh huh...” until “Oh noooooo! Why is this negative?!?!? It ruins everything! Go back and fix it!!!” And then the next student moved forward in the queue.<br /><br />We went through this process of working and my checking and sending kids back for about 40 minutes. “Nooooo!!!” I would circle something and pretend to freak out, sending them back to fix stuff. It became a carnival. “AAACK!" I would say. "Go back and fix this!” Seeing this happen in real-time changed kids’ relationships with their misconceptions. It reminded me of piano practice as a child. Something would splatter in the midst of a phrase or figure and I would have to go back to the beginning, willing the figure to come out right through my fingers on the keyboard.<br /><br />We kept going until all 12 students had triumphed over slope and point-slope form. What I loved about this process was how it transformed the social focus of the process of understanding to us (the whole class) against the skills. Kids were going wild and cheering when somebody finally triumphed over the "slope and point-slope” process, jumping up and down and hugging their newly non-reassessing classmates and friends.<br /><a href="https://1.bp.blogspot.com/-UMFRkcDoJZU/WEBZoi9BDUI/AAAAAAAAA74/XoFPp1VqaPQTgBJ2dofTXjlxh0UgCYRGgCLcB/s1600/20161201%2Bcollective%2Bwin%2Bas%2Ba%2Bteam.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="115" src="https://1.bp.blogspot.com/-UMFRkcDoJZU/WEBZoi9BDUI/AAAAAAAAA74/XoFPp1VqaPQTgBJ2dofTXjlxh0UgCYRGgCLcB/s200/20161201%2Bcollective%2Bwin%2Bas%2Ba%2Bteam.png" width="200" /></a><br />The goal became one of getting everybody in the classroom over the finish line. From individual mastery to collective. Students stopped focusing on who had status in the classroom and who did not. They stopped thinking about their place in the social hierarchy and instead lost themselves in the flow of doing this f***ing slope and linear equation problem.<br /><br />And then as each new classmate finally triumphed over the skill, the whole class felt truly victorious.<br /><br />Everybody got the individualized attention they needed as they drove their skills forward, and we also bonded as a collective community, advancing our work as a group.<br /><br />In these divisive times, I think this might be an important process to cultivate.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-64870963560730737512016-11-22T12:08:00.002-08:002016-11-22T12:08:45.366-08:00Blogs feed my teaching soul, but Twitter helps me feel less aloneAs I've been working on my National Board certification, I have come to appreciate something that the #MTBoS gives to me every single hour of every single day.<br /><br />My tweeps' teaching blogs feed my teaching soul. Seriously, any time I feel stuck, I can just fire off a search in the MTBoS search engine and BOOM, I get drenched in a downpour of borrowed genius.<br /><br />But in the moment, when I am freaking out or feeling lost, Twitter makes me feel ever so much less alone.<br /><br />So thank you, my tweeps, and keep going. I am thankful for you Every. Single. Day.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-38534334324531808422016-10-23T10:12:00.002-07:002016-10-23T17:10:22.722-07:00Graphing Stories Meets Estimation 180: A Love StoryOn Friday we started our Functions unit, and as always, my go-to is to start with Dan Meyer's <a href="http://blog.mrmeyer.com/2007/graphing-stories/">Graphing Stories</a>.<br /><br />After they pick up a Graphing Stories worksheet, I give them a brief set-up, explain that the guy in the video is my friend Dr. Meyer (though now I have to explain that this was a long time ago, in a galaxy far, far away, when he was a young Jedi-in-training), and promise them that I will rewind the videos as many times as they want, to whatever point they want, for as long as they want, so we can figure out as closely as possible what their graphs ought to look like.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-j0YlolXcAi4/WAzzuoT5rvI/AAAAAAAAA7U/kquMFVK3sGk803bTn0MdhS9KUMwWXJFMACLcB/s1600/Graphing%2BStories%2B1.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="screen shot of Dan leaning over the railing of the footbridge" border="0" height="192" src="https://1.bp.blogspot.com/-j0YlolXcAi4/WAzzuoT5rvI/AAAAAAAAA7U/kquMFVK3sGk803bTn0MdhS9KUMwWXJFMACLcB/s320/Graphing%2BStories%2B1.png" title="" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Portrait of the artist as a young Jedi warrior</td></tr></tbody></table>Only this time, we encountered a spontaneous twist.<br /><br />The first video went off without a hitch. Using QuickTime, I have an action-only version of the first 15-second video (Dan walks over a bridge in a Santa Cruz park), so that it doesn't accidentally reveal anything I'm not ready to reveal yet.<br /><br />I explained that my friend Dr. Meyer is unusually tall (I gave his actual height, which can be found on his web site) and this gave students the idea to "measure" him onscreen so they could try to better estimate the rise of the bridge.<br /><br />This gave me an idea.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-t8QiyQXqiDM/WAzy5QS8ahI/AAAAAAAAA7Q/zVLvasdnWrcmTA8-b4ztfd9BgIAOq6i6wCLcB/s1600/graphing%2Bstories%2Bmovie%2B2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="221" src="https://4.bp.blogspot.com/-t8QiyQXqiDM/WAzy5QS8ahI/AAAAAAAAA7Q/zVLvasdnWrcmTA8-b4ztfd9BgIAOq6i6wCLcB/s400/graphing%2Bstories%2Bmovie%2B2.jpg" width="400" /></a></div>The second video in the series is of Dan descending some exterior condo stairs, but after the first viewing, an argument broke out in the discussion as they tried to find some hook they could use to improve their estimations. "A car is, like 6 feet high!" "No, the stairs are about 5 feet in total!" Blah blah blah.<br /><br />Enter <a href="http://www.estimation180.com/">Estimation 180</a> thinking.<br /><br />My classroom is right next to the door to the stairs up to the small faculty parking lot. I pointed to two kids. "OK, you and you — take a yard stick, <i style="font-weight: bold;">carefully</i> cross the driveway, and go measure some average car heights in the teachers' parking lot."<br /><br />The arguments continued so I pointed out another kid, one who was extremely interested in the stair height but who has never before piped up in class. "Pick a helper, grab a yard stick, and go out to the stairs and find an average for the stair height."<br /><br />While our data gatherers performed their missions, the estimation arguments continued inside the classroom. "The car is only about 4 feet high!" "No, it's not!" "What about the lamp post—can we use that?"<br /><br />Five minutes later, our intrepid explorers returned and we harvested our newfound information.<br /><br />Then we continued working on the Graphings Stories task.<br /><br />The funny thing was that our estimates led to wildly wrong answers, but that wasn't the most important thing. Instead, what was powerful was the level of engagement in discovery that electrified the room.<br /><br />Instead of considering the need to quantify to be yet another tedious task that stood in the way of getting "the right answer," students started to lose themselves in the flow of the process of mathematizing their world.<br /><br />And isn't that the whole point?<br /><br />So thank you to Dan (<i>@ddmeyer</i>) and Andrew (<i>@mr_stadel</i>) for giving me the tools to slow my kids down and help them to find the wonder in everyday situations.<br /><br />PS — I still haven't revealed the "answer" to Graphing Story #2 yet. But I'm curious to see what unfolds in class when we come back tomorrow. ;)cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-26056682995038777592016-10-19T10:49:00.000-07:002016-10-19T12:15:16.269-07:00Scaffolding Proof to Cultivate Intellectual Need in GeometryThis year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.<br /><br />I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.<br /><br />We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof. We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.<br /><br />Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.<br /><br />Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."<br /><br />While this seems <i>necessary</i> to me at times and for many students (especially during the early stages), it also seems dramatically <i>insufficient</i> because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."<br /><br />So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).<br /><br />I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.<br /><br />Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.<br /><br />This has led to some amazing discussions of argumentation and logical dependencies.<br /><br />An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:<br /><br />-<a href="https://drive.google.com/open?id=0B8XS5HkHe5eNamZXTndlRlVCSFE">Sample proof to be sequenced & justified</a><br /><br />Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.<br /><br />I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of developing intellectual need in his work!cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com6tag:blogger.com,1999:blog-5779271385256625533.post-54616531619694724932016-10-01T08:40:00.002-07:002016-10-01T08:44:19.958-07:00Algebra 1 inequalities unit - notes on a conceptual and problem-based approachMy Algebra 1 inequalities unit is now the unit with which I am most satisfied, including a solid conceptual launch and framework, some serious problem-based learning, a deep treatment of quantitative reasoning and logic, and an excellent amount of discourse-rich practice activities designed to achieve both procedural <i style="font-weight: bold;">and</i> conceptual fluency, so I want to document here how it works. If there is useful stuff in it for you, then great! But I'm not going to be posting a lot of user-ready materials here to print off and give to students, so I want to be totally up front with you.<br /><br />Inequalities are still one of the most procedurally taught of all Algebra 1 topics (see Holt Algebra 1, CPM, Exeter, or anything else you can find, if you have any doubts about this), and yet, it has seemed to me for a long time that they offer the opportunity for students to ground their new understandings in one of their most accessible, intuitive existing mathematical understandings — namely, their sense of when a quantity is "more than" or "less than" some other quantity.<br /><br />As Nunes and Bryant clearly show, children have a very deep understanding of comparing quantities from a very young age — including both discrete and continuous quantities. For this reason, it has long seemed essential to me to hook Algebra 1 students' work with understanding with their own authentic and meaningful prior knowledge.<br /><br />So in my unit,we start by activating prior knowledge with Talking Points and problem-based learning from the Exeter Math 1 sequence. In the initial part of the unit, we investigate whether quantities are going to be more than or less than other quantities. We go from comparing known, computable quantities (such as one sum or difference to another sum or difference) to more abstract quantities (such as, If x is greater than -1, then x + 2 is greater than 1), always pressing them to rely on their own understanding or constructed understanding.<br /><br />The conversations that were sparked during the initial phase — both this year and last year (with Patrick Callahan in my room) — were fascinating to all.<br /><br />My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together), it has to be a guideline that will help them to ground their <i>new</i> ideas in their <i>old</i> understanding of something. This has led me to come up with much more sensible and conceptually-based guidelines that students both remember and rely on correctly. For example, one of our "rules" is that when you have some kind of inequality, it makes sense to put it into <i>"<b>Number Line Order</b>"</i> — in other words, always organizing statements or representations about smaller quantities to the left of statements or representations about larger quantities. This has had the benefit of getting students to realize that the real number line is a tool that they can use for their own benefit. It is not just another arbitrary math teacher whim.<br /><br />Another "rule" we co-developed as a community was the idea of <b><i>the</i></b> <i><b>subjectivity of x </b>—</i> namely, that in mathematical sentences and statements, we should organize our symbolic representations so that <i>x</i> is the subject of our sentences. Another way that students expressed this idea was that <i>x</i> is the hero or heroine of our story, so <i>x</i> should be the subject of our sentences.<br /><br />This too has had a profound effect on students' understanding of inequalities. Most textbooks emphasize equivalence of the statements 2 > x and x < 2, but from the perspective of student understanding, privileging both statements equally is just stupid. What we are hoping for students is that they will see that as they are trying to represent values of <i>x</i>, that makes <i>x</i> the hero of our story. So our mathematical statements should be organized to reflect the subjective status of <i>x</i> — not the "correct" or "incorrect" equivalence of the statements. This gave students the idea that they have permission to <i>change</i> statements around to suit <i>their own </i> learning, and that they do not need to distort themselves to please whatever mathematical authority decided to place given statements in a form that is backwards for them.<br /><br />We have also developed a "rule" for <b><i>distinguishing absolute value expressions from other groupings in equations and inequalities</i></b>. The idea we have developed is that <b><i>absolute value expressions need to be "isolated" and "unpacked" into all possible cases of ordinary equations or inequalities</i></b> (i.e., NON-absolute value statements) before you can apply any solving or simplifying techniques. Our shorthand for this method is: 1 - ISOLATE, 2 - UNPACK, 3 - SOLVE. Then once you have solved, you should put everything into Number Line Order so you can INTERPRET and GRAPH your findings.<br /><br />This idea of <b><i>unpacking absolute values first into all possible cases</i></b> is a strong way of getting students to <b><i>reason quantitatively and abstractly</i></b> rather than blindly applying rules. Since they have some guidelines that are grounded in their own logical understanding and personal experience, my students have been much more thoughtful and intentional when they approach absolute value inequalities and equations. It has also led to a deepening of students' understanding of when the distributive property can and should be applied to a grouping and when it should not. This has greatly deepened the discussions of the functioning of groupings among students to the better.<br /><br />Because we have build a solid conceptual foundation for our work with absolute values, <b><i>practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets</i></b> from 1 to 35 odd. At every step of the way, we have been using Speed Dating as a discourse-rich activity in which students can apply their understanding to a wide range of different problems of varying levels of complexity. It also demands that they <b><i>share</i></b> their understanding and <i style="font-weight: bold;">speak about it</i> with their classmates, sometimes giving and sometimes receiving assistance.<br /><br />In this way, practice activities have become a new form of conceptual investigation themselves, and are not mere mental exercises designed to enhance procedural efficiency. As I am discovering through my work on National Board Certification, our profession has a major problem with privileging "discovery" activities as the only ones that occasion and showcase conceptual understanding. As How People Learn clearly lays out, it is often the case that students do NOT have a 'Eureka' moment during the initial 'discovery' segment but rather, like the young learners they are, they have a lightbulb moment once they have experienced a problem in multiple different contexts. This leads me to one of my pet peeves about why teacher certification and National Board Certification overvalue only the initial discovery activity as a venue for showcasing student insight, but that is a complaint for a different blog post.<br /><br />Another benefit of treating practice as an opportunity for applied investigation is that it avoids the contrived, dog bandana pseudo-context problems that <i>strive</i> to provide a real-world context, but only end up twisting themselves into senseless knots. The contexts should not be harder for students to make sense of than the absolute value problems themselves! The most accessible real-world absolute value problem we have worked with is the question, When is water NOT a liquid? This makes sense to my 9th graders. Convoluted problems about the conditions under which a value will land within a range of possible values are not helping here, people. I am finding that this is a case where treating absolute value problems as a system of equations or inequalities to be unpacked and investigated as number sense problems has been far more worthwhile — and far more rewarding for students. As Deborah Ball has said, Sometimes mathematics is its own best context.<br /><br /><br />cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com3tag:blogger.com,1999:blog-5779271385256625533.post-19297899842054214362016-09-07T15:22:00.001-07:002016-09-07T15:22:14.505-07:00Katamari and Speed Demons and Not-KnowingI want to keep track of how I am learning from my speed demons and katamari in Algebra 1.<br /><br />I want to be clear about two things. Not all speed demons are alike. And also not all katamari are alike.<br /><br />Slowly I am finding my way to the most discouraged and shut-down learners. I group and regroup, based on formal and informal assessments. Also based on intuition. The great thing about using PBL and the adapted Exeter problem sets is that together they are giving me a lot better information and opportunities to identify, target, and support my students. Most of what we do in class is problem-solving in groups — interpreting, reasoning, and making sense using group whiteboards and whatever other tools we need.<br /><br />I notice the need to provide much more intensive support to most discouraged katamari than I had expected. These are the most shut-down of my learners. These are the students who are being quiet to avoid being noticed. In math class, they live in a defensive psychological crouch. I know this posture well. But the only way I can understand it is to join the process and probe. I ask questions: when you are AT such and such, how far away are you? <br /><br />I am really shocked at how shut down and disassociated these students are from their intuition and their natural intelligence. I wonder how long they have been cultivating this posture of hiding.<br /><br />I draw a crude map on my paper and point to different positions. When you're here *pointing to the diagram*, are you further or closer to X? *moves pencil closer to destination* What about now? what about now?<br /><br />By probing, I discovered something amazing — three tables of students didn't really understand what it meant to be AT the destination.<br /><br />I tried different tactics until finally I asked, when you're IN your kitchen at home, how far away are you from your refrigerator? How many miles away are you? It took a while to convince them that they actually knew they were ZERO miles away.<br /><br />Once they figured out what the meaning of being AT someplace or crossing the destination was/is, they could begin to build a table. One step after the other. Let's work backwards -- when you're one hour away, how many miles away are you?<br /><br />This is a lack of basic trust in their own innate natural functioning. A lack of trust in their own intelligence and problem-solving skill. It takes skill to get this going. They have to be prompted/encouraged to start from where they're at — not to hide the fact that they are lost.<br /><br />These students are good hiders. They know how to hide in plain sight. They are skilled spies. They know how to evade detection. I am more tenacious than most. I do not accept evasion. I probe, I question, I support, and if need be, I help. I help because they are too shut down emotionally to take the risk of humiliation. They are exhausted from not-knowing and from hiding their not-knowing. But not-knowing is the start of knowing. You can't begin to know until you know that you do NOT know.<br /><br />Not-knowing is an empty space in which knowing can arise.<br /><br />Katamari don't realize this yet, but they are closer to finding out. Speed demons have no clue about it yet, and they are so far away from discovering it there's not even any point in raising the issue yet. Katamari are so much closer to the truth. I don't mention any of this to any of them. I just keep probing and asking questions and asking what if and how much and how do you know.<br /><br />At the end of class, one of my speed demons told me she would really like to collaborate (unlike the other speed demons at her table, who are just zooming along and only occasionally conversing). She asked if she could move to Table X, where some very discouraged Ss are. These are the students I was probing the most with. I was thrilled. She was eavesdropping on us when I was working with them. These are the questions she herself likes to ask and think about.<br /><br />I don't have any conclusions to offer, just these noticings and wonderings.<br /><br />We are still in the early community-building days, when discovery is young.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-46602681551525872552016-08-31T17:20:00.002-07:002016-08-31T17:20:31.465-07:00More on katamari and speed demons #MTBoSBlaugustGotta squeak this one in under the #MTBoSBlaugust wire. :)<br /><br />In the world of meditation retreats, what I've come to think of as "Emotional Breakdown Day" of a five-to-seven-day sesshin is pretty much always on a Wednesday (or Day 3, if the retreat doesn't start on a Sunday). You can set your watch by this. There is something about getting psychologically and emotionally heated up that happens after you've been simmering for three days. I think this is true at all spiritual retreats around the world. At a three-day retreat like Twitter Math Camp, it comes at Day 1.5. All of a sudden, Twitter and blogs are flooded with snippets or full-on geysers of despair at how great everybody else seems to be doing and how totally crappy you [INSERT YOUR OWN NAME HERE] are as a teacher.<br /><br />I have tried to learn not to take this seasonality personally, but it's hard not to. When you go deep, you get invested.<br /><br />This cycle seems equally true for me during the school year. Week 3 is inevitably my emotional breakdown week. I can no longer keep up the pace (or the illusion of the pace) and the kids can no longer keep it up either. So things start to break. Students act out. Norms fall apart. I lose my shit.<br /><br />This is just the nature of the cycle of practice. Like winter, or hurricane season, or the World Series, It. Happens. So the real test is how I am going to deal with it.<br /><br />I changed the seating chart in 7th block and rolled out my best rethinking of katamari and speed demon problem-based learning practice (see "<a href="http://cheesemonkeysf.blogspot.com/2016/02/lessons-from-lessons-from-bowen-and.html">Lessons from 'Lessons from Bowen and Darryl</a>'"). I put the speed demons with other speed demons so they would leave my katamari alone already. I want the katamari to learn how to trust their own minds, their own guts, their own hearts. They lack confidence. But put a bunch of them together, and they have no choice but to trust themselves and each other. Without the speed demons to carry them along, they have to think.<br /><br />And I tell you, my friends, it was magical.<br /><br />I revised the day's problem set to put the Important Stuff at the top (though I never label it as Important Stuff — that gives it too much weight for teenagers, plus too little weight for everything else), instructed them to get one table whiteboards, two markers, and a washcloth and I yelled, "GO!" I think the yelling is a particularly artful piece of instructional practice.<br /><br />The room began to hum and glow in the late afternoon fog. This freed me to question and support the groups that felt particularly stuck — to help them get just unstuck enough to keep going.<br /><br />They didn't even care when they worked beyond the time limit I had set for this work.<br /><br />I especially loved the spontaneous alliances that formed across difference. After the first really juicy problem, two young men who hadn't said 'boo' to each other in the first two and a half weeks — a young black student and a Chinese-American student — gave each other a particularly complicated, multi-part handshake than made my heart smile. A table of girls cheered when they finished the same problem.<br /><br />This is why I believe that getting students into a state of flow when they are doing mathematics is the most important thing. If you align yourself with everything we know is good and healthy and whole about doing math, then everything else will proceed smoothly.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com1tag:blogger.com,1999:blog-5779271385256625533.post-17832649118110739092016-08-28T09:45:00.001-07:002016-08-28T16:41:21.672-07:00Week 3 — Talking Points — Desert Island Thinking #MTBoSBlaugustWhen I arrived at Princeton, I had been placed with three other roommates in a Gothic dorm suite. I was proud of the things I had accomplished so far. I'd been tied for valedictorian at a huge and competitive public high school, I'd been a soloist at the All-State choral and orchestral concert, and I'd been president of and/or varsity lettered in all my extracurriculars.<br /><br />So as I discovered that everybody I met had <i style="font-weight: bold;">also</i> been valedictorian, editor of the school newspaper, an All-State varsity athlete or musician, etc., I had quite an adjustment. I had to learn how to stay present with my own inner experience and on what was in front of me directly.<br /><br />I've been thinking about that experience these past two weeks as I have been watched my incoming 9th graders at Lowell adjust to the shock of discovering what it means to arrive at the next level.<br /><br />The classes are much, much more demanding than they are used to, even at the strongest middle schools. And in addition, as every visitor can see at the front entrance of our school, there is a board that celebrates accomplishments of many of our Lowell graduates since our founding in 1856. There are three Nobel laureates, Pulitzer Prize winners, Broadway and Hollywood stars, admirals and generals and and politicians and world leaders. There are sports legends and pop culture icons, civil rights heroes, the founder of The Gap, and friggin' Lemony Snicket, among others.<br /><br />Every race, ethnic background, and gender seems to be well-represented.<br /><br />No wonder my poor kids are freaking out.<br /><br />Now I try to imagine what it must be like to be one of the very few African-American students in our school. Some of them appear to be doing just fine, but I imagine it is a very strange and disorienting experience to find yourself in what must seem like an endless ocean of whiteness.<br /><br />We are trying to be intentional in how we are supporting these students and transforming our school culture. We are following best practices and reflecting critically on how we are doing and how we can support their experience. I wish that I could magically airlift in a larger number of faculty of color so that they felt more reflected in the adult community they see all around them. But that is not how public education works. And we have no time to indulge in magical thinking.<br /><br />So this is the point at which I am introducing some Talking Points on what I like to call desert island thinking. It is the best way I have found to help students to cope with their own feelings of imposter syndrome and the need to be their own best supporters as they enter a completely new territory.<br /><br />I call it desert island thinking because it is what helped me to cope when I felt overwhelmed and alone as a freshman at Princeton. I reminded myself over and over that, if I were stuck on a desert island, I would want to be with other smart and motivated and hopefully good-hearted people because that would give us our best chances to survive and thrive.<br /><br />In my teaching life, I think of this as Otter Nation. Our motto is, Hold hands and stick together. When sea otters sleep, they hold hands so they don't drift apart from their tribe. The same is true of us math teachers. We hold hands through the #MTBoS and through #educolor , through Twitter and blogs, and through every social media-based method we can find.<br /><br />We hold our students in our hearts and try to give them every possible support and advantage we can provide.<br /><br />For me, a part of that involves helping them to become metacognitively aware and and self-reflective about what they are experiencing and how they can cope with it, how they are brave and well-equipped and the advantages of holding hands and sticking together.<br /><br />Our Latin Club has hoodies with one of my favorite lines from Virgil's <i>Aeneid</i> emblazoned on the back: forsan et haec olim meminisse iuvabit, which I would loosely translate as, "Perhaps some day we will laugh about this." This is pretty much where many of my students — and especially my students of color — find themselves at the start of Week 3 too. At this point in Aeneid I, Aeneas and his troops have been driven from their homeland in Troy and find themselves on storm-tossed seas, wondering how they are going to survive.<br /><br />A growth mindset, and some desert island thinking, along with Talking Points about it, are the best support I can offer them.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-_uaczhVPXC8/V8MU0By3uUI/AAAAAAAAA6Q/fhWhhKt442QO2yDCZ6I1_W4gleJVypRzACLcB/s1600/20160828%2BTalking%2BPoints%2B-%2Bdesert%2Bisland%2Bthinking.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://4.bp.blogspot.com/-_uaczhVPXC8/V8MU0By3uUI/AAAAAAAAA6Q/fhWhhKt442QO2yDCZ6I1_W4gleJVypRzACLcB/s640/20160828%2BTalking%2BPoints%2B-%2Bdesert%2Bisland%2Bthinking.png" width="492" /></a></div><br />cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com1tag:blogger.com,1999:blog-5779271385256625533.post-69461002530544468262016-08-20T11:24:00.003-07:002016-08-20T11:29:38.527-07:00Week 1 - "very much like being shot out of a cannon" #MTBoSBlaugustWeek 1 is in the can and I wanted to blog one of my best ideas from my first week back.<br /><br /><a href="https://4.bp.blogspot.com/-3GsiXOQIauM/V7iewJG0sMI/AAAAAAAAA48/5vaEpxejEXglt7oktC9XxC79z5sNuv-ugCLcB/s1600/201608%2Broom%2Bset%2Bfor%2B39%2Bstudents.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://4.bp.blogspot.com/-3GsiXOQIauM/V7iewJG0sMI/AAAAAAAAA48/5vaEpxejEXglt7oktC9XxC79z5sNuv-ugCLcB/s200/201608%2Broom%2Bset%2Bfor%2B39%2Bstudents.jpg" width="200" /></a>I should start out by saying that Week 1 was very much like being shot out of a cannon — much more so than usual. My classes this year are huge — 36, 37, or 38 students per class — but my room is the smallest in the school. So it took a lot of clever angling and arranging to ensure that we could have enough desks in the room and that everybody could more or less see from their given position in the room. I always mark on the floor with a Sharpie so that it's easier to put the desk clusters back into their optimal positions. Once upon a time, I would have considered this a form of vandalism, but now...? Hey, that's just common sense.<br /><br /><div style="text-align: left;"></div><a href="https://3.bp.blogspot.com/-pMtDmZAJ8fI/V7ifzNjHIRI/AAAAAAAAA5M/G8Y7XDeWSGwLJJS0tdNBy79ElH4ETP7YwCLcB/s1600/Graffle%2Bseating%2Bcharts%2Bidea.graffle.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="154" src="https://3.bp.blogspot.com/-pMtDmZAJ8fI/V7ifzNjHIRI/AAAAAAAAA5M/G8Y7XDeWSGwLJJS0tdNBy79ElH4ETP7YwCLcB/s200/Graffle%2Bseating%2Bcharts%2Bidea.graffle.jpg" width="200" /></a><br /><b>SEATING CHART MANAGEMENT</b>I had a major conceptual breakthrough with seating charts this August. I always use OmniGraffle to set up my basic seating chart/management chart template that I use on my clipboard to take attendance and make notes. This year, it occurred to me: instead of using those stupid little name card tags with names to make a wall chart (which takes up an unreasonable amount of time), why not just make a board with a sheet of vinyl across to hold blank copies of the week's seating charts?<br /><br /><div style="text-align: center;"><b><i><visualize added="" be="" chart="" dazzling="" here="" holder="" monday.="" my="" of="" on="" photo="" that="" the="" will=""></visualize></i></b></div>So now, I can just print off two copies of my updated charts — one for my clipboard and one for the wall pockets. And voilá! Easy to change the seats around.<br /><br />The stapled blob of paper charts makes it super-easy to make notes about collaboration or mathematical successes during group work. It also makes it easy to enter attendance on the computer each class because I just look for the 0s or 1s. My scribbled comments make it easy to enter "Professionalism" scores or comments, or to send e-mails to students or families. My working copy of charts gets stapled together and placed on my clipboard. When the week is over, I archive it in a big binder.<br /><br /><div style="text-align: center;"><b><i>[visualize the dazzling photo of my wall chart that will be posted here on Monday]</i></b></div><br />There's a lot more to say about Week 1, but I'm still recuperating. More soon!<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-CYJzwhHv8XM/V7ifEGVqw2I/AAAAAAAAA5A/IRgKh-AbZ5ogkImvpx0JFsgMkyx8biOLQCLcB/s1600/handouts%2Bhanger.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://2.bp.blogspot.com/-CYJzwhHv8XM/V7ifEGVqw2I/AAAAAAAAA5A/IRgKh-AbZ5ogkImvpx0JFsgMkyx8biOLQCLcB/s200/handouts%2Bhanger.jpg" width="150" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Hey, Megan, Here's my <br />handouts hanger <i>in situ</i>!</td></tr></tbody></table>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-72591972213606772582016-08-12T06:26:00.000-07:002016-08-12T06:26:12.360-07:00#MTBoSBlaugust 4 - Dear Pam (starting the year and Exeter Math 1)Dear Pam,<br /><br />The closer the start of school gets, the more I realize I need to be wise about how I choose to introduce Exeter problems. I want to set my students up for success. They are an extremely diverse group, coming from all over the city and all over the world. They are new to our huge school and to high school. And my classes are huge. There is still load-balancing to do, but right now my Algebra 1 rosters are topping out at 42. This is a special problem in my room because we can only fit 36 desk-chairs in the room. Literally. So I have to practice basic trust that this is all getting worked out through placement and class balancing and other dark arts of administration.<br /><br />But all of this thinking has made me realize that I need to rely on my proven, first-two-weeks training plan to get them ready for my course and to help accomplish all of the transitions I am expecting them to manage. Bottom line: I need to start by training them with Talking Points to ensure equity and access for all.<br /><br />Accordingly, I've created a set of <a href="https://drive.google.com/open?id=0B8XS5HkHe5eNMG95cm04cWNGZkk" target="_blank">Day 1 Talking Points about ratios, units, and rates</a> (actually for us it's Day 2 because Day 1 is a loss), which are the topics of M1:1#1. I want to use this to train them on norms and practices in my class so that we can begin our problem-based learning together. For me, the first two weeks are critical. They are when I train my students intensively on my rules and norms, using Talking Points and other practices. With 36 students per class, this is a must. Everybody has to get bought in and skilled at the structures so we can proceed.<br /><br />I also need to work this beginning-of-the-year routine into my four-stage How People Learn groove. Talking Points is my go-to beginning-of-the-year Stage 1 task structure (initial encounter with new concepts and activation of old). Our debrief and shift into doing notes/INB organization is Stage 2 (initial provision of a new expert model). We'll do some notes on ratios, units, and rates to review and organize our thinking, and then we'll do a little combo-plate Stage 3 / Stage 4 work (deliberate practice with metacognitive / transfer task) in the form of some opening Exeter problems.<br /><br />I've changed my mind about ordering page 1. I'm going to have them do 1#1 first (easy application of ratios, units, and rates) as a "deliberate practice with metacognitive awareness" problem and then 1#5, the journey of a thousand miles problem (J-1000). What I like about this one as a first transfer task is that it pushes them to use their bodies and to involve their whole selves in the investigation. Once somebody in the class asks, "Hey, Dr. S — can we have a yardstick?" I know I've got them. ;)<br /><br />From here we can move on to 1#2, which is really the heart of the intro to problem-based learning in my opinion.<br /><br />But realistically, we won't get to this until Day 3 (which is my true Day 2). So I'm having to accept that this is the reality of where I can get them and when. I can tweak the rest of everything to make it fit the first two weeks, but this is my reality.<br /><br />So Day 3 needs a new set of Talking Points to begin and then a handout for problem 1#2, the problem about counting non-stop by ones to one billion.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-47779171147706287832016-08-05T13:15:00.000-07:002016-08-05T13:25:29.925-07:00#MTBoSBlaugust 3 - The Bumper Car Theory of Anti-Racist Training for Teachers and StaffThis one is challenging to write because I want to honor all appropriate boundaries while inquiring into my own personal experience of the process.<br /><br />This next week, during our whole-school PD on Wednesday, we are embarking on our first year of a multi-year program of anti-racist training for teachers and staff. Earlier this summer, I was one of 25 teachers and staff from our school who attended the initial training, and naturally, nothing went as planned. Does it ever? Heavy Sigh. So this morning, we did our reset and met about our plan to do this training with our whole school.<br /><br />The enterprise of confronting privilege to teach and learn about privilege is daunting, and it is unavoidable that many people who encounter this work will quickly get rubbed raw. In some ways, that is by design. You can't remain comfortable while digging into uncomfortable territory. But at the same time, conceiving the work merely as a project of "disruption" dishonors the good will and long-term focus of individuals who have come together on their own out of their own deep-rooted belief that we need to do better, both for our students and for ourselves.<br /><br />So you can see how it's a complicated and messy process to get started.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-_0h9WLkJuTk/V6T1LN81y5I/AAAAAAAAA4M/UR3mWIohVWA-5yirnaU8Cl9JiSajT2GngCLcB/s1600/wile-e-coyote-help.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://4.bp.blogspot.com/-_0h9WLkJuTk/V6T1LN81y5I/AAAAAAAAA4M/UR3mWIohVWA-5yirnaU8Cl9JiSajT2GngCLcB/s200/wile-e-coyote-help.png" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Your face here</td></tr></tbody></table>What I am coming to understand about it all is this: in order to have courageous conversations about race, we need to learn how to see our own personal invisible beliefs. These are hard to see because they are by nature invisible. They are blind spots. For example, as a high-status, highly educated white teacher, I tend to feel confident in sharing my views publicly; but at the same time, I struggle to keep my passion and confidence from appearing as arrogance to those with different patterns of privilege. And I'm just one individual teacher in a very large faculty. I'm sure that other teachers struggle to notice their own patterns of privilege. Plus the nature of the dominant culture in our school is unusual and complex. So all in all, learning to see the individual and collective belief systems and blind spots is going to be a real challenge. We are going to need to spend a lot of time in a space of collective and individual not-knowing, together. And I fully expect that process to be uncomfortable.<br /><br /><a href="https://3.bp.blogspot.com/-Vop4ZneL51Y/V6T137BNXnI/AAAAAAAAA4Q/BwAjOCoHMz4mBFIrR4oxUicXr3hc2ha4gCLcB/s1600/bumper%2Bcars.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" src="https://3.bp.blogspot.com/-Vop4ZneL51Y/V6T137BNXnI/AAAAAAAAA4Q/BwAjOCoHMz4mBFIrR4oxUicXr3hc2ha4gCLcB/s200/bumper%2Bcars.jpg" width="200" /></a>What strikes me most is that this whole process is like being in racial identity bumper cars. Like at a carnival. We need to <b><i>expect</i></b> to be disturbed and surprised and confused as we discover how other drivers in their own identity bumper cars interpret and experience life from their own points of view, because everybody is so certain that their own personal bumper car point of view is clear-seeing and constructive and intentional. But every time the ride starts up, whenever you try to steer your own bumper car, you cannot help but crash into other people's bumper cars. So the process of investigation is complicated because there is no way to step outside of the bumper car bumping arena while the inquiry is ongoing.<br /><br /><div style="text-align: left;"></div>From the 30,000-foot perspective, I can see that the bumper car system is designed to thwart objectivity. In their own bumper car ride, nobody is 100% in control of their own bumper car. We all have our own projections and privilege and beliefs that we project onto every other driver who crashes into us. If you consider how the bumper cars are designed, you may understand logically that the bumping is unavoidable. But after you've been in the arena for a little while, trying to steer your own car for a bit, it becomes hard <b><i>not</i></b> to take things personally. It becomes impossible to avoid lapsing into the belief that other drivers are intentionally crashing into you to push you off course.<br /><br />I think this model is especially true when you've got a large room full of public school educators — smart, highly educated, open-hearted people who do what they do out of dedication to learning and to contributing to the common good. The moment you start to prod individual teachers into seeing how they benefit from various networks of privilege, things get painful. People shut down or break down. And I've never yet seen it handled well. In our culture, teaching is already pre-constructed as a "Wretched of the Earth"-level of profession. Poorly paid, micro-managed, and bullied by corporate reformers and unelected politicians. What could <b><i>possibly</i></b> go wrong when you try to confront public school teachers about privilege?<br /><br />So I think it is going to take a certain gentleness, determination, and persistence to help a whole faculty to see how we as individuals benefit from different forms and degrees of privilege, both in our school culture and in our society. It is also going to take chocolate and a whole lot of radically appropriate self-care. I am hopeful in the long term that we will make progress, but I suspect that in the near term, things could get messy. Still, I remain optimistic and curious to see how things unfold.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com2tag:blogger.com,1999:blog-5779271385256625533.post-43793876735056185242016-08-03T17:21:00.003-07:002016-08-03T17:28:16.955-07:00#MTBoSBlaugust Post #2: Using Exeter Math 1 — down to brass tacksI am shamelessly using the #MTBoSBlaugust challenge as a prod to organize my thinking about how to use Exeter Math 1 this year in Algebra 1.<br /><br />Last year's Algebra 1 implementation was a mess. Not even a hot mess, just a mess. The materials from our district were inadequate and even with my enrichments, I found that the whole was not coherent enough or challenging enough for my students. It also started waaaaaaayyyy too slow out of the gate. And it didn't provide nearly as much work on modeling and sense-making as I wanted.<br /><br />Other than that, it was fine.<br /><br />So this year, to start with, here's my game plan for reordering units for the fall semester.<br /><br /><b>Sequence of Units</b><br /><ol><li>Mathematical Modeling and Problem-Solving <span style="color: #45818e;">(<i><b>EQ: </b>How can we use mathematics and logic to make sense of real-world situations?) </i></span></li><li>Equations, Proportions, and Number Lines <span style="color: #45818e;">(<i><b>EQ: </b>How can we use our existing algebra toolkit to solve equations involving proportions, variables, and absolute value?) </i></span></li><li>Lines and Linear Functions <span style="color: #45818e;">(</span><i><span style="color: #45818e;"><b>EQ: </b>How do lines and linear functions enable us to analyze and predict real-world phenomena?)</span> </i></li><li>Functions and Functional Thinking <span style="color: #45818e;">(<i><b>EQ: </b>How can functions and functional thinking help us in our modeling work?) </i></span></li><li>Systems of Linear Equations <span style="color: #45818e;">(<i><b>EQ: </b>How can we use multiple equations to model real-world situations?) </i></span></li><li>Working With Exponents <span style="color: #45818e;">(</span><i><span style="color: #45818e;"><b>EQ: </b>What makes exponents such powerful tools?)</span> </i></li></ol>Unit 1 is going to be my two-week mathematical modeling and problem-solving boot camp.<br /><br /><b>Begin At The Beginning: Unit 1</b><br />This is going to work because M1:1 – 8 cover all of the things I want to deal with as review topics anyway, but in a novel and challenging way: rates and units; "micro-" modeling (going from words to mathematics) and functional thinking; using the number line; modeling with non-standard rates, fractions, ratios, and proportions; distributive property and order of operations; "like" terms; and modeling with area and volume.<br /><br />Meanwhile, the most important thing about the first two weeks is to really drive home my routines and norms and structures. That creates space to get to know students and do some formative assessment so I can make intentional groupings and establish the tone for the course.<br /><br />This year, the material for the first 9 days will be the first eight pages of lightly adapted Exeter problems, i.e., replacing "Exeter" with my school's name and replacing all the kid names with character names of my own choosing (I tend to favor Batman and the character names from Sesame Street and Harry Potter).<br /><br />Day 1 is a complete loss because (a) it's a ridiculously short period and (b) we are required to go over our syllabus (don't ask). But I think I will put M1:1#5, which I refer to as J-1000 (the "journey of a thousand miles" problem) on the opening slide as the intro task for students to do once they've found their seats. It's a little bit of math, but it sets the tone I want, which is that we get down to mathematical business in my classroom.<br /><br />Day 2 is my true Day 1, so that will be my M1:1 day.<br /><br /><b>Background: Organizing Principles</b><br />For any day that I am using a page of Exeter problems, I want to organize the problems according to Bowen and Darryl's PCMI-based framework of Important Stuff / Interesting Stuff / Tough Stuff.<br /><br />That way, I can also use their twin strategies of (a) intentional groupings (keep the speed demons away from the katamari) and (b) deliberately featuring katamari solutions and insights during whole-class discussion segments. For background on all of this, including PCMI, Bowen, and Darryl's group work strategies, first read <a href="https://researchinpractice.wordpress.com/2016/01/28/lessons-from-bowen-and-darryl/" target="_blank">Ben Blum-Smith's Lessons from Bowen and Darryl</a> and then my post on <a href="http://cheesemonkeysf.blogspot.com/2016/02/lessons-from-lessons-from-bowen-and.html" target="_blank">Lessons from Lessons from Bowen and Darryl</a>.<br /><b><br /></b>Any remaining problems left over from the day's classwork can be done as homework problems that night, and those can then be discussed during the next day's <a href="http://cheesemonkeysf.blogspot.com/2015/12/burning-questions.html" target="_blank">Home Enjoyment/Burning Questions segment</a>. A beautiful thing.<br /><br /><b>Day 1: M1:1#2</b><br />Problem 1:#2 is a perfect rich task for starting off an Algebra 1 course. You can demand that students let go of their habits of learned helplessness and use whatever they know. You can encourage them with problem-solving process hints and without robbing them of the opportunity to do the thinking for themselves.<br /><br />Everything else on page 1 is just review and activation of prior knowledge.<br /><br />OK, that's all I've got for page 1.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com8tag:blogger.com,1999:blog-5779271385256625533.post-63287279681998059632016-08-01T13:56:00.004-07:002016-08-01T13:58:28.198-07:00Scan in your best worked examples #MTBoSBlaugustThis is going to be a short post because I am trying to take my own best advice:<br /><br /><blockquote class="tr_bq">Scan in your best worked examples.</blockquote>Let me repeat that: Scan in your best worked examples. Scan in your best worked examples.<br /><br />And if you are just joining us... today's advice is: scan in your best worked examples.<br /><br />I spent about 2 months working through every problem on every page of Exeter Math 1. All 91 pages of problem sets. And finally, today is ScanFest 2016.<br /><br />I first learned this advice from Sam Shah (of course), and it bears repeating. Your frazzled, middle-of-the-year self will thank you for it.<br /><br />And now, back to scanning.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com3tag:blogger.com,1999:blog-5779271385256625533.post-69971576238364383132016-06-23T17:36:00.001-07:002016-06-23T17:40:37.937-07:00Exeter Math 1 Reflection 3: A Course in Advanced Proportional Reasoning<i><span style="color: #76a5af;">This is the third in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.</span></i><br /><div></div><br />As I see it, there are two core developmental strands in Exeter Math 1 that are woven together throughout the course. One strand concerns advanced proportional reasoning. The other involves what I would characterize as Exeter's method of "micro-modeling"—an ongoing spiral of frequent, small, subtle modeling tasks that provide extensive both variety and depth of practice in modeling. Many variations are explored so that students get a lot of practice in making sense of similar and differing contexts.<br /><br />What I love about this blend of proportional reasoning and micro-modeling is that it occurs at the intersection of advanced textual interpretation and advanced proportional reasoning. This means it is an immersive experience in relentless sense-making and meaning-making as students explore modeling. In this course, mathematical modeling is a full-contact sport. Having worked all the way through the entire course, I can see how it is going to develop great fluency and confidence in modeling for Algebra 1 students, regardless of where they are starting (assuming, of course, that they have the basic prerequisites for Algebra 1 success).<br /><br />The opening problem sets are deceptively simple, although page 1 problem 2 (from here on out, I'm going to use the Exeter-style notation of 1#2 to mean "page 1 problem #2), would be a fantastic Day 1 in-class rich task that drops students right into a hard micro-modeling problem with whatever tools they have.<br /><br />But other than 1#2, most of the problems in the first 7 pages are deceptively simple. They're clearly written to review prior knowledge and to establish individual and group norms of work, with the major themes being work on rates, distributive property, order of operations, functional thinking, notation, number line, negatives and opposites, #<i>unitchat</i>, fractions, reciprocals, and rational numbers. Major concept development focuses on distance = rate - time, distributive property, working with various kinds of graphs and graphical representations, and micro-modeling.<br /><br />And then you arrive at 8#1, and BLAMMO.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c9MHImR-Xp4/V2x-545OeyI/AAAAAAAAA30/BwPaKZNOdScxIvW9WGVtdOzDZ2PC-O7LACLcB/s1600/Exeter%2BMath%2B1%2Bpage%2B8%2Bproblem%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="164" src="https://2.bp.blogspot.com/-c9MHImR-Xp4/V2x-545OeyI/AAAAAAAAA30/BwPaKZNOdScxIvW9WGVtdOzDZ2PC-O7LACLcB/s640/Exeter%2BMath%2B1%2Bpage%2B8%2Bproblem%2B1.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">This is what I'm thinking of when I talk about a truly rich task blast-off.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I'm not going to give away the punch line here, but this "box within a box" problem is an excellent example of what I mean when I say the course focuses on advanced proportional reasoning. The problem requires a very advanced analysis of many distinct moving parts, along with an ability to track back and explain your thinking. By my count, this problem requires the learner to navigate and articulate issues of area, volume, footprint, a difference of footprints, layering, and negative space. Perhaps you can see other ideas here as well.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So if you're just getting started with Math 1 and wondering what the heck all the fuss is about, I encourage you to hang in there. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">I imagine that we will get to page 8 around the middle to end of the second week of school. And when we do, students should know that the fun is just beginning. :)</div>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com0tag:blogger.com,1999:blog-5779271385256625533.post-78696513944078830582016-06-22T09:51:00.002-07:002016-06-22T10:01:57.720-07:00Exeter Math 1 Reflection 2: Growing Up as a Mathematical Thinker<i><span style="color: #76a5af;">This is the second in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.</span></i><br /><div><br /></div><div>First and foremost, Exeter Math 1 is a course in growing up as a mathematical thinker. It is about leveling the playing field between and among rising 9th grade students. </div><div><br /></div><div>Here's how I would would frame this journey for students: This is a course in developing your own mathematical self-reliance and resourcefulness as a learner. Your Essential Question is always: <i>How can I exhaust everything I already know before I ask the teacher for help?</i></div><div><br /></div><div>You already know an enormous amount of mathematics. In this class, you will need to step forward with that and be willing to attack problems with the best thinking you already have. You may not know everything, but you always know <i style="font-weight: bold;">something</i>, and since that something is the best thing you know, you show up and start there and give it everything you've got.</div><div><br /></div><div>Then, when you have struggled as much as you can and as hard as you can—both by yourself and with your table group—and when you can no longer do anything more with what you've got, <i style="font-weight: bold;">that</i> is the appropriate point at which you can ask the teacher for help.</div><div><br /></div><div>That is the best use of the teacher.</div><div><br /></div><div>If you are passive in this work or mess around, you are going to suffer.</div><div><br /></div><div>This course works at two levels. At the content level, we are going to do all of the usual content work in an Algebra 1 class. <i>But the more important work we will do always takes place at a metacognitive level. It is designed to help you learn how <b>you</b> learn advanced mathematics</i>.</div><div>____________________________________________________</div><div><br /></div><div>OK, back to the teacher perspective.</div><div><br /></div><div>I have a Post-It on the inside-front cover of my binder on which I wrote this:</div><blockquote class="tr_bq">Exeter discovery is about guided sequential flailing.</blockquote> I think this is true. The Exeter Math 1 path definitely involves guided, well-sequenced flailing. It also integrates continual spiraling designed to activate prior knowledge. The purpose is always to discover how much math you already know and can put into service with the problems that are directly in front of you.<br /><br />There is mathematical content and metacognitive content on each page.<br /><br />This leads to the issue of practice. In Exeter Math 1, there is a very specific theory of action in the practice problems that are given and in how they are used. There are none of the usual taking-up-time, too-easy practice problems. If students need extra practice on certain specific procedures, then you have to source them yourself from someplace else, such as (for us) the Holt Algebra 1 textbook.<br /><br />But that is OK because at this point in my career, I can do that in my sleep.<br /><br />The Exeter Math 1 approach to practice problems is to provide juicy, meaningful, gimmick-free practice problems that are (a) always of medium difficulty or above and (b) integrated with metacognitive reflection and discussion. For this reason, I would be inclined to use these inflection points in the curriculum as opportunities to use Talking Points to solidify conceptual understanding and to get students exploring and articulating the subtle misconceptions and potential pitfalls inherent in practice problems of a medium level of difficulty or above.<br /><br />This is a very deep teaching idea to me — to keep practice problems at or above a medium level of difficulty and to have students explore and give voice to these subtleties as rich opportunities to make meaning in their work.<br /><br />More thoughts coming soon.cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com3tag:blogger.com,1999:blog-5779271385256625533.post-206333931191734672016-06-21T13:23:00.000-07:002016-06-21T14:13:38.857-07:00First thoughts on completing Exeter Math 1<br /><div>I just finished doing the 2010 edition of Math 1 (91pages) today. Now begins the synthesizing and summarizing, which I will put into blog posts.</div><div><br /></div><div>Math 1 is an Algebra 1 course that includes an incredibly deep coverage of proportional reasoning, in addition to the usual linear, quadratic, and exponential function topics.</div><div><br /></div><div>I did Math 1 because most of our incoming students are incredibly bright and hard-working but they were not the math monsters in their middle schools. They have many of the typical middle school gaps, but they are much more sophisticated than most 9th grade Algebra 1 students. So the fact that Math 1 is a REALLY TOUGH course that dives very deep into Algebra 1 material is a great thing because it will give my students the deep rich course they deserve, even though they are placed into Algebra 1 based on their current skill level.</div><div><br /></div><div>My Algebra 1 learners find themselves stuck in a ZPD no-man's-land: their ZPD as math learners is nowhere near their ZPD as readers. </div><div><br /></div><div>This presents a huge problem in the classroom. The math in CPM Algebra 1, for example, is rich and interesting, but the text is written for reluctant readers, discouraged readers, and English Language Learners, which is a huge turn-off for the vast majority of my enthusiastic and highly capable readers.</div><div><br /></div><div>They feel insulted by it, and they are not shy about expressing these feelings. So my student population tends to dismiss it and resist it, even if they really do need to learn the content. This raises the question of how best to serve a population of learners who need to be challenged with greater nuance in textual interpretation and presentation in an introductory high school math class.</div><div><br /></div><div>For all of these reasons, Math 1 is going to form a terrific problem-based “spine” for my Algebra 1 classes. The problem sequences are rich and interesting and engaging with sophisticated contexts, though they start from first principles. They develop to a point where even a mathematically sophisticated adult will find them very challenging.</div><div><br /></div><div>To get started, I printed all pages of the problem sets, answer keys, and commentaries and created a binder with the following sections:</div><div><br /></div><div>1 - Problem Sets plus glossary at the end</div><div><br /></div><div>2 - Commentaries</div><div><br /></div><div>3 - My Worked Solutions (for each page of problems, I have one stapled cluster of my worked solution pages)</div><div><br /></div><div>4 - Answer Keys</div><div><br /></div><div>I did all of my work on three-hole binder paper, with each new page from the problem set being its own stapled packet (or "blob") in the Worked Solutions section. </div><div><br /></div><div>Whatever problem set I was working on I would take out of the binder along with the relevant answer key page. That way I could work on binder paper without having to carry the whole damn binder around all the time. Much of this work was done on a lap desk with my iPhone/Desmos for graphing, my TI-83-plus (sorry, Eli) for computation, and my monkey pencil case including my mechanical pencil, my ProRadian protractor, and my colored pencils.</div><div><br /></div><div>A lot of people have asked me why I started at the end and worked from the end forwards, about 10 pages at a time. The answer has two parts: (1) whenever I started from the beginning, I bogged down or got sidetracked; and (2) it enabled me to see where we were going and where students would end up. By seeing where they would land at the end of the course, I could better understand how things worked from the beginning.</div><div><br /></div><div>More thoughts coming soon, but I wanted to capture these ideas right away. If you have specific questions you'd like to discuss, please put them into the comments section below.</div>cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com6tag:blogger.com,1999:blog-5779271385256625533.post-22175292240567545652016-06-18T13:26:00.001-07:002016-06-22T10:02:36.690-07:00CompostingIn addition to our school vegetable garden and internal CSA (Community Supported Agriculture program), we have a coop with 8 laying hens. The AP Environmental Science classes work in the garden every Monday and harvest the produce that will be provided to faculty and staff subscribers during the school year. In this way, the garden program is self-funding, and students learn from experience exactly how difficult it is to grow food to meet expectations.<br /><br />The chickens in the garden are our program's local celebrities. The kids love the chickens and the fresh, local chicken eggs are a very prized item for CSA subscribers.<br /><br />The chickens, however, are a little bitchy. Bitchy and not too bright.<br /><br />The reason we know this is that the faculty have to take care of the chickens once the school year is over and the kids are out of school.<br /><br />Taking care of the garden is one thing. It's very meditative to come in to the garden once a week and pull weeds in the sunshine (or more likely, in the fog). Taking care of the chickens, on the other hand, can be a pain. As I said, they are bitchy and often peck at their caregivers as well as at each other. They knock over their feeding bins, poop in their communal water trough, and hide their eggs away from sight.<br /><br />For this reason, it's always a good idea to tend our chickens on the buddy system.<br /><br />So during our weeklong training session at school this week, I often brought my lunch down to the garden with my friend and colleague Cathy Christensen to hang out in the rare sunshine and assist with the chickens. One of the other AP Environmental Sciences teachers, Kathy Melvin, was already there between the rows of kale and chard, pulling weeds and listening to Bob Dylan.<br /><br />Cathy and I ate our lunch, bravely fought off the chickens as we righted the water bins, filled up their food stocks, and gathered eggs.<br /><br />When we were done, I asked Kathy M whether I should put my compostable fork into the compost pile with my plate and napkin. As a new composter, I am sensitive to the limits of low-tech composting systems, as opposed to our city's industrial strength composting system. I was trying to be mindful.<br /><br />She said, "It won't decompose in there, but it will spark a good conversation."<br /><br />That answer told me everything I need to remember about why I love our school and my colleagues there. As Michelangelo wrote to himself in his 80s (on a note found in his studio after his death), "Ancora imparo — I am still learning."cheesemonkeysfhttp://www.blogger.com/profile/09311170815422010013noreply@blogger.com1