tag:blogger.com,1999:blog-47194656734395252252017-06-21T19:41:35.328-07:00Mindful About Math EducationIdeas from research and practice for teachers and learners of mathematics, particularly in the middle grades.EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-4719465673439525225.post-5108394159992368862015-03-23T14:15:00.003-07:002015-04-08T07:07:34.743-07:00Challenging Math TopicsI'm writing a series of blog posts about challenging topics in K-8 math standards. I'll post the links to these here as they come out.<br /><br />March 16, 2015 <a href="http://gettingsmart.com/2015/03/spotlight-on-math-strategies-for-addressing-the-most-challenging-math-standards/" target="_blank">Overview of Challenging Standards</a><br /><br />March 23, 2015 <a href="http://gettingsmart.com/2015/03/spotlight-on-math-strategies-for-addressing-the-most-challenging-math-standards-measurement/" target="_blank">Measurement</a><br /><br />March 30, 2015 <a href="http://gettingsmart.com/2015/03/spotlight-on-math-strategies-for-addressing-the-most-challenging-math-standards-modeling/" target="_blank">Models with Mathematics</a><br /><br />April 6, 2015 <a href="http://gettingsmart.com/2015/04/spotlight-on-math-strategies-for-addressing-the-most-challenging-math-standards-fractions/" target="_blank">Fractions</a>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com6tag:blogger.com,1999:blog-4719465673439525225.post-40893686820782234062014-09-04T09:18:00.000-07:002014-09-04T09:18:04.617-07:00Growth Mindset in Teaching and Learning Mathematics<span style="font-family: Georgia, Times New Roman, serif;">Since I first<a href="http://foundmath.blogspot.com/2010/04/thinking-about-ability-fixed-vs-growth.html" target="_blank"> posted about mindset research in 2010</a> there has been much discussion and application of this in relation to mathematics teaching and learning so I figured I'd update things with new references. One of the most impressive efforts is that of mathematics teacher Helen Hindle who has created <a href="http://www.growthmindsetmaths.com/" target="_blank">Growth Mindset Mathematics</a> that's chock full of great information and lesson activities! Jo Boaler, mathematics educator at Stanford, has written extensively <a href="http://www.ncpdf.org/pdf/steering/2013-09-06/12.0%20Boaler_FORUM_55_1_web.pdf" target="_blank">about mindset and mathematics</a> and has an <a href="https://class.stanford.edu/courses/Education/EDUC115-S/Spring2014/about" target="_blank">online course about mathematics learning</a> built around the idea of having a growth mindset. And the research base has continued to grow including <a href="http://blogs.kqed.org/mindshift/2014/07/new-research-students-benefit-from-learning-that-intelligence-is-not-fixed/" target="_blank">this 2014 study</a> showing positive impacts on student achievement after a 30-minute online intervention. Finally, this TEDx Talk by Eduardo Briceno about Growth Mindset gives a great framing of this concept and how it can powerfully drive learning. </span><br /><span style="font-family: Georgia, Times New Roman, serif;"><br /></span><div class="separator" style="clear: both; text-align: center;"><object width="320" height="266" class="BLOGGER-youtube-video" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0" data-thumbnail-src="https://ytimg.googleusercontent.com/vi/pN34FNbOKXc/0.jpg"><param name="movie" value="https://youtube.googleapis.com/v/pN34FNbOKXc&source=uds" /><param name="bgcolor" value="#FFFFFF" /><param name="allowFullScreen" value="true" /><embed width="320" height="266" src="https://youtube.googleapis.com/v/pN34FNbOKXc&source=uds" type="application/x-shockwave-flash" allowfullscreen="true"></embed></object></div>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com7tag:blogger.com,1999:blog-4719465673439525225.post-70826079899912300612014-07-23T10:52:00.001-07:002014-07-24T09:49:44.975-07:00Misconceptions in Mathematics and Diagnostic Teaching<span style="font-family: 'Trebuchet MS', sans-serif;">With the adoption of new standards for learning mathematics that place an emphasis on building students' conceptual understanding and using that to inform their development of procedural knowledge, there is a need for teachers to develop skills in uncovering students' thinking and providing opportunities to address misconceptions. This article by Elizabeth Green in the New York Times offers a thorough look at the changes being called for: "<a href="http://nyti.ms/1nnhIcV" target="_blank">Why Do Americans Stink at Math?</a>" </span><br /><span style="font-family: 'Trebuchet MS', sans-serif;"><br /></span><span style="font-family: 'Trebuchet MS', sans-serif;">The following set of resources offer some good ideas to start approaching mathematics teaching conceptually by eliciting and building from students' thinking:</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://math.serpmedia.org/diagnostic_teaching/" target="_blank">Diagnostic Teaching for Grades 6 and 7</a> (set of 12 problems and accompanying teacher materials)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://www.apa.org/education/k12/student-thinking.aspx" target="_blank">How Do My Students Think? Diagnosing Student Thinking</a> (research-based guide for teachers)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://www.tes.co.uk/teaching-resource/Professional-Development-in-Maths-6093160/" target="_blank">Learning from Mistakes and Misconceptions PD</a> (slides and guide)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://www.annery-kiln.eu/gaps-misconceptions/" target="_blank">Gaps and Misconceptions in Maths</a> (focuses on subtraction, division, and fractions)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://www.sagepub.com/upm-data/37758_chap_1_tobey.pdf" target="_blank">Mathematics Assessment Probes</a> (first chapter of <i><a href="http://www.amazon.com/Uncovering-Student-Thinking-Mathematics-Grades/dp/1412980550" target="_blank">Uncovering Student Thinking in Mathematics, Grades K-5</a></i>)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://www.sagepub.com/books/Book238090#tabview=samples" target="_blank"><i>Uncovering Student Thinking About Mathematics in the Common Core, Grades 6-12</i></a> (free sample chapters)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://iccams-maths.org/CSMS/" target="_blank">Chelsea Diagnostic Mathematics Tests</a> (scanned versions of a book of diagnostic assessment from 1984)</span><br /><span style="font-family: Trebuchet MS, sans-serif;"><br /></span><span style="font-family: Trebuchet MS, sans-serif;"><a href="http://elicitingmathematicalmisconceptions.wordpress.com/" target="_blank">Eliciting Mathematics Misconceptions</a> (blog with two videos about fraction misconceptions)</span>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com4tag:blogger.com,1999:blog-4719465673439525225.post-54556915012306349402014-05-13T08:32:00.001-07:002014-05-13T08:32:10.106-07:00Ideas for Mathematics Education CoursesThe ideas in the spreadsheet below came from conversations with mathematics education faculty in the California State University system. It's far from a comprehensive list but is at least a start! You can <a href="https://docs.google.com/forms/d/1lSd9XoqUhJDSVIZoYqTdR4A_a6ZhJ_0OfEtGVA1vhwg/viewform" target="_blank">add your ideas using this form</a>.<br /><br /><br /><iframe frameborder="0" height="600" src="https://docs.google.com/spreadsheet/pub?key=0AoINcPpRuRo6dDk2WFZzS05JbWhiY0puVTR4dkkwTGc&single=true&gid=0&output=html" width="600"></iframe>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com2tag:blogger.com,1999:blog-4719465673439525225.post-82028826719239486772014-04-14T08:00:00.001-07:002014-04-14T08:01:31.798-07:00Technology that Supports Meaningful Mathematics Learning<span style="font-family: Trebuchet MS, sans-serif;">At the NCTM Annual Meeting in New Orleans last week I had the chance to talk briefly with a few folks about their use of technology that supports meaningful mathematics learning. Some criteria we agreed were important to keep in mind when deciding whether to use technology and, if so, which technology, included:</span><br /><ul><li><span style="font-family: Trebuchet MS, sans-serif; line-height: 1.15; white-space: pre-wrap;">Helps students see things differently (within constraints - e.g. microworlds)</span></li><li><span style="font-family: Trebuchet MS, sans-serif; line-height: 1.15; white-space: pre-wrap;">Is simple to use but promotes deep cognition/challenge</span></li><li><span style="font-family: Trebuchet MS, sans-serif; line-height: 1.15; white-space: pre-wrap;">Gives students better access to concepts (reduce/remove barriers)</span></li><li><span style="font-family: Trebuchet MS, sans-serif; line-height: 1.15; white-space: pre-wrap;">Allows students to connect mathematics to aspects of their culture/community</span></li><li><span style="font-family: Trebuchet MS, sans-serif; line-height: 1.15; white-space: pre-wrap;">Puts students into the role of creators rather than consumers of knowledge</span></li></ul><span style="font-family: Trebuchet MS, sans-serif;">Resources that reflect some number of these criteria were shared in a Google Form, the results of which are below. These represent the specific interests of the group I was talking with and the list is of course not comprehensive. If you'd like to contribute additional resources, <a href="https://docs.google.com/forms/d/1xj7R6VA5ladGImnzx7eja7TJm9uN7q4hk7-c8mztpjY/viewform" target="_blank">the form is here</a>.</span><br /><br /><iframe frameborder="0" height="300" src="https://docs.google.com/spreadsheet/pub?key=0AmiQpP2O7pyndDc3MDZLWHdDbXVXV0lDV2cwNVJkTVE&single=true&gid=0&output=html&widget=true" width="600"></iframe> <br /><br />EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com6tag:blogger.com,1999:blog-4719465673439525225.post-86058705701165232242014-02-14T15:37:00.002-08:002014-02-20T13:56:33.338-08:00Piloting Smarter Balanced in California in Spring 2014<span style="font-size: xx-small;">I was asked recently by some colleagues for details about California's piloting of the Smarter Balanced assessment in spring 2014 and thought others might find the information I compiled helpful. Feel free to add comments/clarifications. And if you spot any outright misstatements, certainly let me know!</span><br /><br /><div class="MsoNormal"><span style="color: #1f497d; font-family: Calibri, sans-serif;">As best I understand it, the Field Test in California in spring 2014 will be only computer-based and non-adaptive (each student gets a fixed set of items) for the purpose of testing out item validity and reliability. From the 2015-16 school year, the online assessments will be adaptive (with a student’s response on one item determining what item they get next). Also, from 2015-16 for a total of three years there will be a paper-based version of the assessment (non-adaptive with scores somehow equated to the online, adaptive tests) for schools that lack the technology to conduct the test online. There is a <a href="http://www.cde.ca.gov/ta/tg/sa/sbac-itr-index.asp" target="_blank">Technology Readiness Tool</a> each district is supposed to use to gauge their capacity to support students’ access to the assessments: <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="color: #1f497d; font-family: Calibri, sans-serif;">Students will either use computers with a mouse or other point-and-click device OR a tablet device with touchscreen. Each device must have the <a href="http://sbac.portal.airast.org/browsers" target="_blank">Smarter Balanced secure browser</a> installed to be used for testing. Specific technology requirements are found here (along with <a href="http://www.cde.ca.gov/ta/tg/sa/smarterfieldtest.asp" target="_blank">other field test information</a>).<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="color: #1f497d; font-family: Calibri, sans-serif;">Students are permitted to have access to certain tools, both embedded within the secure browser, and non-embedded. This includes scratch paper on which to do mathematical work and, from grade 6, graph paper. Any scratch work is to be discarded after the test and is not used for scoring purposes. The <a href="http://www.smarterbalanced.org/wordpress/wp-content/uploads/2013/09/SmarterBalanced_Guidelines_091113.pdf" target="_blank">full list of tools is here</a> but note each state may make modifications to this.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><span style="color: #1f497d; font-family: Calibri, sans-serif;">This <a href="http://www.politico.com/story/2013/12/common-core-testing-problems-seem-inevitable-101568.html" target="_blank">Politico storyfrom Dec 2013</a> details some of the challenges with online assessments. And <a href="http://www.stats.indiana.edu/maptools/ISTEPinterruptions.html" target="_blank">this interactive graphic</a> tells the story of Indiana’s experience with online tests last spring (led by McGraw Hill, not Pearson as in CA).</span>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com2tag:blogger.com,1999:blog-4719465673439525225.post-82936666027571709042012-10-11T06:43:00.000-07:002014-05-14T08:59:27.499-07:00NCTM Dallas Regional PresentationThis is a quick post to share the slides from my presentation today on Common Core Standards for Mathematics Practice at the NCTM Regional Conference in Dallas. I'll update later with more details...<br /><br />The presentation slides are <a href="https://app.box.com/s/6ci83s3x0ezieqlctgdy" target="_blank">at this link</a>.<br /><br />Update (12:45 PM 10/11/12)<br />The session was well-attended with about 100 folks in the room. I invited them to enter my <a href="http://www.socrative.com/" target="_blank">Socrative Teacher</a> classroom by installing the Socrative Student Clicker app before we got started. Over 30 participants did so and were able to provide input in addition to those who spoke out verbally during the session.<br /><br />I was impressed with the turnout and the participation. My intent was to provide food for thought about the Common Core Standards for Mathematical Practice as well as what it means, in general, to think about teaching 21st century learners. I hope the attendees found it to be 60 minutes well-spent!<br /><br />One slide I didn't have as much time to comment on as I had planned is titled "Flipping Out" and features a link to a Kahn Academy video. The point I was going to make is that if "flipping" one's classroom is simply a matter of moving lectures to homework and doing homework in class, we're not addressing the real need to transform what teaching and learning are all about. It's more than rearranging the same traditional lessons that focus primarily on skills and procedures. We need to be looking for ways to shift teaching and learning to place students' reasoning and sense making at the center of all that takes place. Two recent columns discuss this quite well (better than I am able to do!), so I encourage you to check these out:<br /><br /><ul><li>NCTM President Linda Gojak, "<a href="http://www.nctm.org/about/content.aspx?id=34585" target="_blank">To Flip or Not to Flip: That is NOT the Question</a>"</li><li>Teacher and Blogger Shelly Wright, "<a href="http://plpnetwork.com/2012/10/08/flip-love-affair/" target="_blank">Why I Gave Up Flipped Instruction</a>"</li></ul>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com3tag:blogger.com,1999:blog-4719465673439525225.post-79351985118460362942012-10-02T09:29:00.000-07:002013-02-04T09:23:59.961-08:00Teachers Who Focus on Student ThinkingOne of my Master's students (a first-year math teacher) recently mentioned how daunting it is to think about shifting one's classroom away from teacher-led lecture and rule/procedure following to an approach that promotes and builds on student thinking generated through inquiry and problem solving. One of her concerns is that only a minority of teachers seems to be working toward such a shift so it would be hard to find folks with whom to collaborate. In addition to connecting her with some local teachers I know who are doing such work, I decided to search for blogs that offer examples of teachers sharing their successes and challenges as they make such shift as a way to encourage my student (and others) to jump in. Here is a sampling of what's out there. If you find the ideas useful, be sure to engage with the bloggers/teachers to let them know and make a connection. You're definitely not alone in this transition!<br /><ul><li>The <a href="http://picrust.wordpress.com/" target="_blank">Pi Crust blog</a> offers ideas and insights from an experienced middle school math teacher, Ms. Krasnow. She's got a ton of great activities and strategies that engage students in reasoning and developing understanding.</li><li>Middle school math teacher Fawn Nguyen's <a href="http://fawnnguyen.com/" target="_blank">Finding Ways to Nguyen Students Over blog</a> offers an awesome set of tasks, perspectives, and good humor for teachers wanting to share ideas for engaging students in meaningful learning.</li><li>An official <a href="http://blog.keycurriculum.com/2012/05/saying-yes-to-thinking-math-problem-solving-and-teacher-response/" target="_blank">blogger for Key Curriculum Press, Karen Greenhaus</a> shares links to videos and other resources that support teachers in eliciting and valuing student thinking. The "What do we 'no'?" video is worth watching and reflecting on!</li><li>The <a href="http://www.teachingtweaks.com/wordpress/" target="_blank">Teaching Tweaks blog</a> is created by a former middle school math teacher and current UCLA doctoral student Belinda Thompson. Her "say this not that" set of entries provide food for thought.</li><li>6th grade math teacher Kirsten Silverman has started her <a href="http://klsilverman.wordpress.com/" target="_blank">Numbers blog</a> to share her work toward making her classroom a place where students engage in reasoning and sense-making. Several good ideas already posted as well as honest commentary on the challenges encountered along the way.</li><li>High school math teacher Amy Gruen's <a href="http://squarerootofnegativeoneteachmath.blogspot.com/" target="_blank">Square Root of Negative One blog</a> has great ideas for supporting students in taking ownership of their (collective) learning. Her entries on having students make up word problems and efforts to train students to be "coaches" for each other are fabulous.</li><li>John Golden, a math professor at Grand Valley State University, has the <a href="http://mathhombre.blogspot.com/" target="_blank">Math Hombre blog </a>where he shares ideas for classroom activities as well as teacher professional development. His <a href="http://mathhombre.blogspot.com/2012/09/put-me-in-coach.html" target="_blank">entry on the use of cognitive coaching between colleagues as a way to improve teacher practice</a> is outstanding and worth you taking some time to read and watch. Also, don't miss his <a href="http://mathhombre.blogspot.com/p/geogebra.html" target="_blank">Geogebra activity links</a>!</li><li>Joe Ochiltree's <a href="http://brainopennow.wordpress.com/" target="_blank">Brain Open Now blog </a>has some great algebraic reasoning activities and insights into how to set these up.</li><li>The <a href="http://inquiryblog.wordpress.com/" target="_blank">Inquiry Blog</a> was started by two teachers, one Australian and one from the United States, who wanted to create a space for teachers working to create an inquiry-based learning environment could share ideas. Interesting, honest, informative.</li><li>Finally, certainly not to be missed, former high school math teacher <a href="http://blog.mrmeyer.com/" target="_blank">Dan Meyer's blog </a>is always full of ideas that challenge traditional notions of teaching mathematics.</li></ul>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com2tag:blogger.com,1999:blog-4719465673439525225.post-57645285423263415222012-09-28T15:35:00.001-07:002012-09-28T15:35:40.980-07:00Strategies for Engaging Students in Mathematics LearningThis presentation was created through a collaboration with a middle school teacher, Ms. Jenny Kim, who works with 7th graders in Norwalk, CA. Though we put this together in 2010, I think the ideas are still quite relevant and powerful to consider. Note that Ms. Kim's students made HUGE gains in their mathematics achievement in the span of one academic year in her classroom with 2/3 moving from below proficiency to at or above proficiency. Hope you find in this some ideas for your own classroom!<br /><br /> <object width="480" height="360"><param name="movie" value="https://voicethread.com/book.swf?b=2114910"></param><param name="wmode" value="transparent"></param><embed src="https://voicethread.com/book.swf?b=2114910" type="application/x-shockwave-flash" wmode="transparent" width="480" height="360"></embed></object>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com5tag:blogger.com,1999:blog-4719465673439525225.post-68174170776920021772012-09-25T09:29:00.004-07:002013-11-14T07:48:38.291-08:00Apps for Math Learning<span style="font-family: Trebuchet MS, sans-serif;">As we see more students having access to web-enabled handheld devices such as tablets it is important that we (the adults in their lives) help them to use these devices productively. Recent research has revealed a <a href="http://www.nytimes.com/2012/05/30/us/new-digital-divide-seen-in-wasting-time-online.html" target="_blank">new "digital divide" though not in access to devices but in knowing how to use the devices to support learning</a>. (Read <a href="http://www.gse.uci.edu/person/warschauer_m/docs/equity.pdf" target="_blank">Warschauer and Matuchiniak's detailed summary of this research</a> to learn more). Of course, this requires us as educators to know a bit more about such tools! With this in mind, here are some places to begin.</span><br /><div><br /></div><div><span style="font-family: Trebuchet MS, sans-serif;">A growing number of apps support meaningful mathematics learning. By this I mean these engage users in more than factual recall or procedural drills (which are okay if that is your objective - for example <a href="http://illuminations.nctm.org/Activity.aspx?id=3563" target="_blank">Concentration </a>and <a href="http://illuminations.nctm.org/pickapath" target="_blank">Pick-a-Path</a> are great!). A few of my favorites are:</span></div><div><br /><ul><li><a href="http://illuminations.nctm.org/Activity.aspx?id=3528" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">Okta's Rescue</a><span style="font-family: 'Trebuchet MS', sans-serif;"> is for preK-2 children and helps them develop number sense by counting, subitizing (recognizing units), and recognizing sets of 10 (when counting how many Oktas were saved). To learn more about the importance of subitizing to students' early development of number, </span><a href="http://teacherweb.com/WA/nachesvalleyprimaryschool/msclark/Subs.PDF" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">read this <i>Teaching Children Mathematics </i>article by Doug Clements</a><span style="font-family: Trebuchet MS, sans-serif;">.</span></li><li><a href="http://motionmathgames.com/motion-math-wings/" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">Motion Math's Wings app</a><span style="font-family: 'Trebuchet MS', sans-serif;"> (only for Apple devices) offers a fun way to learn about multiplication through multiple representations including grouping and array models. A nice article about what underlies students' understanding of multiplication, see </span><a href="http://assets.pearsonschool.com/asset_mgr/legacy/200746/InvestigationsMulitplication_3706_1.pdf" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">Smith & Smith's "Assessing Elementary Understanding of Multiplication."</a></li><li><span style="font-family: 'Trebuchet MS', sans-serif;">Finally, here are a couple of apps to develop fraction understanding. <a href="https://itunes.apple.com/us/app/mathtappers-estimate-fractions/id353282053?mt=8" target="_blank">Math Tappers: Estimate Fractions</a></span><span style="font-family: 'Trebuchet MS', sans-serif;"> asks you estimate where a fraction or sum of two fractions would be placed on a number line and how a fraction is represented visually. It's a simple app done quite well. NCTM offers <a href="http://illuminations.nctm.org/Activity.aspx?id=3510" target="_blank">Equivalent Fractions</a> for </span><a href="http://itunes.apple.com/us/app/equivalent-fractions/id548051011" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">Apple </a><span style="font-family: 'Trebuchet MS', sans-serif;">and </span><a href="https://play.google.com/store/apps/details?id=air.com.learningtoday.equivalentFractions" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">Android </a><span style="font-family: 'Trebuchet MS', sans-serif;">devices. It allows users to develop understanding of the relationships between visual representations of a fraction and the placement of the fraction on a number line. See </span><a href="http://foundmath.blogspot.com/2010/10/teaching-and-learning-fraction-concepts.html" style="font-family: "Trebuchet MS", sans-serif;" target="_blank">my earlier blog entry about fraction learning</a><span style="font-family: 'Trebuchet MS', sans-serif;"> to get a sense of how to support students in developing understanding of this critical set of ideas.</span></li></ul></div><div><span style="font-family: Trebuchet MS, sans-serif;">Also note that while the apps shared here are for mobile devices, there is a treasure-trove of web-based applets that support conceptual learning of mathematics on <a href="http://illuminations.nctm.org/" target="_blank">NCTM's Illuminations site</a>! If you've not been there for a while, check out what's new.</span><br /><span style="font-family: Trebuchet MS;"></span><br /><span style="font-family: Trebuchet MS;">Update 9/28/12 Edutopia released today a free (with registration) guide, <span style="color: #1f497d; font-family: "Calibri","sans-serif"; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-themecolor: dark2;"><span style="color: black;"><em><a href="http://www.edutopia.org/mobile-devices-learning-resource-guide" target="_blank">Mobile Devices for Learning:What You Need to Know</a></em>. </span></span></span></div>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com4tag:blogger.com,1999:blog-4719465673439525225.post-67145054523531927242012-08-31T09:26:00.001-07:002012-08-31T09:26:14.274-07:00Social Bookmarking for TeachersI've recently learned about <a href="http://www.diigo.com/" target="_blank">Diigo</a>, a social bookmarking tool with some cool features that include highlighting pages and pasting sticky notes. These get saved to your Diigo account so that no matter what computer or device you're using, once you login to Diigo your bookmarks, highlights, and comments are there. Even better is the ease with which you can share web pages with others, Diigo users and non-users alike. In addition to linking to your Facebook, Twitter, or Google+ account, you can also send a simple URL to folks like this: <a href="http://diigo.com/0so2w">http://diigo.com/0so2w</a><br /><br />If you're a teacher, Diigo is offering a free upgrade to an educator account that allows you to form groups and "enroll" students with their own accounts. Anything that any of the students bookmarks is then shared with the entire group. This is a great way to tap into students' inclination toward social media.<br /><br />And, to top it off, as a Diigo Educator, you'll get a cool badge to put on your site or blog!!<br /><br /><a href="http://www.diigo.com/profile/mwellis" title="mwellis"><img alt="diigo education pioneer" src="http://www.diigo.com/teacher_entry/badge?name=mwellis&token=80aafe112e6612c37a116a16ed1e53df&num=2" style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;" /></a>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com4tag:blogger.com,1999:blog-4719465673439525225.post-77241371509191281822012-05-12T09:09:00.000-07:002012-05-12T09:12:22.908-07:00Sessions at NCTM's Annual MeetingIf you weren't able to attend NCTM's Annual Meeting this year in Philadelphia, you can catch the highlights and get speaker handouts online! My own session, <a href="http://nctm.confex.com/nctm/2012AM/webprogram/Session9331.html" style="background-color: white; color: #633100; font-family: Verdana, Geneva, Arial, Helvetica, sans-serif; font-size: 11px; font-weight: bold; text-align: left;">Eliciting Mathematical Reasoning with Digital Tools: Engaging Students and Teachers</a>, was part of the Learn-Reflect strand focusing on the use of technology in mathematics education. (Note the slides I uploaded to share don't have the embedded multimedia that I used live to keep the file size reasonable). One of the more exciting parts of my session (at least for me) was being able to use the <a href="http://www.socrative.com/" target="_blank">Socrative Teacher and Socrative Student Clicker apps</a> (free on Android and iOS) to allow participants to send in responses. I had about 1/3 of the folks there who could install the app and join in. With a large group, it was a nice way to get feedback from more than just the handful of folks who might speak up orally. This can be really rich to use in your own classroom to promote sharing ideas in a very non-threatening way. You can set it up to accept multiple choice or open text responses.<br /><br />You can search the entire <a href="http://nctm.confex.com/nctm/2012AM/webprogram/" target="_blank">conference program here</a> to look for handouts from other sessions that might be of interest. And don't miss the<a href="http://www.nctm.org/conferences/content.aspx?id=33201" target="_blank"> Opening Session keynote by Diane Ravitch</a> that NCTM has webcast on its site. She was inspiring and basically issued a call to action for us to work even harder to take back our profession from what she termed the "corporate reform" of public education that is narrowing the curriculum and limiting our role as educators to make decisions about what is best educationally for the students and communities we serve.EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com4tag:blogger.com,1999:blog-4719465673439525225.post-54876057226975811642011-07-11T14:19:00.000-07:002011-07-11T14:19:53.102-07:00Rote Memorization versus Conceptual LearningA colleague teaching mathematics at a community college recently asked if I knew of research about "the effectiveness of mass practice in terms of long-term memory." Apologies in advance if you don't have access to an academic journal database and can only view the article abstracts.<br /><br />There is research about so-called "human calculators" who do mental math with large sets of figures; in every instance these folks have developed their talent by spending countless hours memorizing arithmetic facts and relationships. Seems in the most extreme cases their brains are wired to support this sort of activity.<br /><a href="http://rstb.royalsocietypublishing.org/content/364/1522/1385.short">http://rstb.royalsocietypublishing.org/content/364/1522/1385.short</a> <br />However, if we are talking about whether mass memorization equates to meaningful learning for the typical person, evidence from the 1950s demonstrated this was not the case beyond the retention of simple facts:<br /><a href="http://www.sciencedirect.com/science/article/pii/S0022101507632996">http://www.sciencedirect.com/science/article/pii/S0022101507632996</a> <br /><br />Learning with connections is far better. Think about early school mathematics books (in the 1700s) that had page after page of formulas to memorize related to commercial transactions involving proportional relationships among different units. The time and effort it took to try to memorize these was inordinate (and drove many away from mathematics). More "modern" textbooks by the early 1900s had students learn about proportional reasoning as a general relationship that could be applied to an infinite number of specific situations - thus by learning one conceptual relationship (which could then be memorized and also re-constructed if learned with understanding), students had a much better chance of both long-term recall and correct application.<br /><br />Some recent articles related to this idea demonstrate quite powerfully the advantages to conceptual learning.<br /><a href="http://onlinelibrary.wiley.com/doi/10.1002/ddrr.45/pdf">http://onlinelibrary.wiley.com/doi/10.1002/ddrr.45/pdf</a> <br /><a href="http://www.academicjournals.org/ERR/PDF/Pdf%202011/Jan/Narli.pdf">http://www.academicjournals.org/ERR/PDF/Pdf%202011/Jan/Narli.pdf</a>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com3tag:blogger.com,1999:blog-4719465673439525225.post-80101889356893471702011-06-20T17:10:00.000-07:002011-06-20T17:10:49.505-07:00Technology in Mathematics EducationThis week Education Week featured an <a href="http://www.edweek.org/ew/articles/2011/06/15/35mm-math.h30.html?tkn=WRCFqWzno1rBnVjtg1MI7OvyvaFnpJJ/Cx9G&cmp=clp-sb-teacher">article about technology in mathematics education</a> where I was quoted a couple of times. The gist of what I had said was the while technology is crucial in supporting mathematics learning, it is not in and of itself a substitute for quality teaching and curriculum materials. Something not mentioned in the article (which I understand does not have room for everything) was my concern that teachers are not being given enough time to for professional development to not only learn about new technology tools and resources but also to learn how to design and implement learning activities with these.<br /><br />In case you have the time and interest, here are a few articles and tutorials worth exploring about exciting tools to support mathematics teaching and learning!<br /><br /><ul><li>Article titled, "<a href="http://www.nctm.org/eresources/view_article.asp?article_id=6431">Investigating Algebra with Virtual Manipulatives</a>" from NCTM</li><li>Tutorials on using the <a href="http://enlvm.usu.edu/ma/nav/doc/index.jsp#flashtutorials">National Library of Virtual Manipulatives</a></li><li><a href="http://mathandmultimedia.com/2011/01/01/geogebra-essentials-series/">Geogebra Tutorials</a> (25 items) from Mathematics and Multimedia blog</li><li><a href="http://www.onemathematicalcat.org/Math/Geometry_obj/GeoGebra_tutorial/intro_to_GeoGebra.htm">Interactive Geogebra Tutorials</a> (and geometry lessons!) from Dr. Carol Fisher</li><li><a href="http://www.geogebra.org/book/intro-en/">Online book about using Geogebra</a> by geogebra.org</li><li><a href="http://www.wolframalpha.com/screencast/introducingwolframalpha.html">Wolfram Alpha</a> intro video (amazing tool if you've not tried it)</li><li><a href="http://www.screenr.com/">ScreenR </a>allows you to record and narrate anything from your screen - easy to use!</li></ul>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com2tag:blogger.com,1999:blog-4719465673439525225.post-88597231287185457802010-11-18T18:23:00.001-08:002010-11-18T18:38:08.695-08:00Ignite event at CMC-S Math ConferenceSo what could I do to "ignite" thinking about mathematics education given the constraints of using 20 slides and just 5 minutes (with the slides auto-advancing every 15 seconds)? This was the challenge I faced as November 6 approached and I prepared for a so-called <a href="http://ignite.oreilly.com/">Ignite event</a>. The result? See for yourself at <a href="http://sineofthetimes.keypress.com/2010/11/harry-plotter-and-the-deathly-shallows/#more-605">Key Curriclum's Sine of the Times blog</a>. Be sure to check out the other presenter's ideas, too, on <a href="http://www.youtube.com/user/KeyCurriculumPress">You Tube</a>. I am sure you'll find something to ignite your thinking about mathematics teaching and learning!EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com1tag:blogger.com,1999:blog-4719465673439525225.post-91295504914904951072010-10-17T10:19:00.000-07:002010-10-17T17:12:48.934-07:00Teaching and Learning Fraction Concepts and OperationsThe Institute of Education Studies (IES), part of the U.S. Department of Education, has recently released a research-based 90-page report titled, "<a href="http://ies.ed.gov/ncee/wwc/pdf/practiceguides/fractions_pg_093010.pdf">Developing Effective Fractions Instruction for Kindergarten through 8th Grade</a>." The report begins by detailing the poor level of understanding of fraction concepts and skills among U.S. students. For example data from international and national exams have shown that "50% of 8th-graders could not order three fractions from least to greatest...[and] fewer than 30% of 17-year-olds correctly translated 0.029 as 29/1000" (p. 6). <div><br /></div><div>A panel of experts reviewed dozens of studies and identified five (5) recommendations for improving fraction instruction. From page 1 of the document:</div><div><div><ol><li>Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts.</li><li>Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.</li><li>Help students understand why procedures for computations with fractions make sense.</li><li>Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems.</li><li>Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.</li></ol></div></div><div>This list fits well with the approaches I've found successful in my own classroom (previously) and in the experiences of my credential students in their classrooms. Learning to make sense of fraction concepts requires deliberate activities that allow for sense-making to occur. </div><div><br /></div><div>Some resources to help you design such learning activities are:</div><div><ul><li><a href="http://www.visualfractions.com">Visual Fractions</a></li><li><a href="http://dww.ed.gov/see/?T_ID=20&P_ID=48#cluster-3">Doing What Works: Fractions</a></li><li><a href="http://www2.edc.org/mistm/">Dozens of Fraction Activities</a></li><li><a href="http://www.uen.org/Lessonplan/preview.cgi?LPid=23394">Modeling Multiplication and Division of Fractions</a></li></ul></div><div>Have fun with these! It is always great to see students "get it" when working with fractions.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com3tag:blogger.com,1999:blog-4719465673439525225.post-82191343962576594702010-09-05T19:14:00.000-07:002010-09-05T19:29:29.489-07:00The Importance of QuestioningFor the past few years I've taught methods courses for both beginning and experienced teachers of mathematics. One element of these courses has been the use of videos of teachers in the classroom to prompt discussion of specific aspects of practice. The lesson captured in a set of <a href="http://www.mmmproject.org/ls/mainframe.htm">video clips here, "Looking for Squares" by Lisa Brown, </a>is one of the best examples I've found of a teacher's use of questioning to guide students' exploration and eventual sense-making of a mathematical concept, in this case square numbers and square root. It might be helpful to read "<a href="http://coe.ksu.edu/allen/Math/RENEW/RENEW%202/Questioning.pdf">Questioning our Patterns of Questioning</a>" by Herbel-Eisenmann and Brefogel, to provide a context for thinking about the use of questioning in a mathematics classroom. What specific strategies and teaching skills does Ms. Brown use to promote, sustain, and engage student thinking? How does this support their sense-making?EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com1tag:blogger.com,1999:blog-4719465673439525225.post-40909043755614485682010-05-09T18:25:00.000-07:002010-05-09T19:03:27.405-07:00Content Knowlege vs. Pedagogical Content KnowledgeSince its introduction by educational psycholigist Lee Shulman in the late 1980s, the term "<a href="http://www.leeshulman.net/domains-pedagogical-content-knowledge.html">pedagogical content knowledge</a>" (PCK) has been used often to refer to the particular knowledge needed by teachers to effectively create learning environments that support student sense-making in mathematics (and other disciplines). However, until recently there has been no good way to measure a teacher's PCK. <br /><br />This is no longer the case and we now need to learn from the work that has led to greater insights into PCK among teachers of mathematics. With the recent work of U. S. mathematics education researchers <a href="http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf">Heather Hill and Deborah Ball </a>some progress has been made with this at the elementary and middle school levels and researchers led by <a href="http://tsg.icme11.org/document/get/397">Jurgen Baumert </a>in Germany have helped to capture PCK among secondary teachers of matheamtics. The bottom line is that while content knowledge is clearly necessary for teachers of mathematics, it is not sufficient. The most effective teachers also have insights into the mathematics content their students will learn that allows them to identify misconceptions, set up lessons in which students make connections, and provide students with questions and prompts that help them access important concepts.EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com1tag:blogger.com,1999:blog-4719465673439525225.post-50356347572542482372010-04-19T07:30:00.000-07:002010-04-19T09:32:32.628-07:00Thinking about Ability - Fixed vs Growth MindsetIn the U.S. many folks perceive there to be a "math ability" that one either has or does not have (<a href="http://www.springerlink.com/content/pt715k7m38683888/">see Uttal's research</a>). This actually does much harm to how we structure mathematics learning and limits achievement among students. As a teacher I constantly sought ways to engage students in making sense of mathematical concepts and relationships and believe that with few exceptions the majority of people can learn the mathematics in the K-12 curriculum. This was shown to be true among my students, many of whom went from believing mathematics to be a black hole of nonsensical symbols and rules to realizing the logical structure of mathematics and developing the power to reason mathematically to solve problems. This was possible in large part because I felt they were capable of making significant improvements to their knowledge of mathematics.<br /><br />Important <a href="http://www.stanfordalumni.org/news/magazine/2007/marapr/features/dweck.html">research from Carol Dweck and her colleagues at Stanford</a> has shed light on the relationship between one's "mindset" toward ability (mathematical or otherwise) and the actions to which this leads in terms of opportunities to learn and feedback to learners. Ultimately, these have serious consequences on learning outcomes. The idea is this: if one believes ability to be fixed - a fixed mindset - this will lead to actions that a) serve to identify who is high-ability and not and b) provide feedback that reinforces ability status, directly and indirectly. The result - a few students are "smart" in mathematics while many others are not. The smart ones must strive to retain that labeling through "looking smart" (often at any cost) while the others see no reason to work at learning what they are being told is beyond their ability. The result is that actual learning among all students suffers.<br /><br />What <a href="http://www.npr.org/templates/story/story.php?storyId=7406521">Dweck and colleagues found is that changing this script can have pronounced effects</a>. Taking a "growth mindset" that views ability as derived more from effort than some innate quality leads to very difference choices about learning environment and feedback to learners. If one believes most all students can learn mathematics, those who are struggling to do so are in need of support. Feedback such as, "if you work hard at this you will improve" leads to greater motivation and, as their research has shown, greater success. For students who do well with mathematics, the feedback in the growth mindset tells them, "you made a good effort at this and have done well." This is critical because when the mathematics does get challenging (and it will!), these students will persist, putting in more effort, rather than look for ways to simply maintain their "looking smart" status (e.g., shortcuts, cheating, or bowing out).<br /><br /><a href="http://www.brainology.us/webnav/whatismindset.aspx">See here for more about this work and suggestions for teachers and parents.</a>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com1tag:blogger.com,1999:blog-4719465673439525225.post-66361931428704142682010-04-11T16:58:00.001-07:002010-04-11T17:26:04.655-07:00Getting Ready for NCTMWhy would 10,000 teachers of mathematics descend on San Diego for a week? When the annual <a href="http://www.nctm.org/">NCTM </a>Conference is being held there! Check out the amazing list of sessions <a href="http://www.nctm.org/sandiego">here</a>.<br /><br />As a member of the planning committee, I will be in charge of making sure things run smoothly with sessions in the Marriott, so be sure to stop by and say, "Hi!" if you're there.EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com1tag:blogger.com,1999:blog-4719465673439525225.post-20824135388931365062010-04-05T18:24:00.000-07:002010-04-05T18:31:05.679-07:00Welcome!Welcome to my new foundational-level mathematics blog! I plan to share resources and perspectives on the teaching of mathematics at the middle school and early high school level from my experience as a teacher and math educator. My philosophy of teaching is to create a learning environment in which students are encouraged and supported in making sense of mathematics. My work in public Title I schools in California with learners of all backgrounds convinced me that students have the potential to understand mathematics and the our work as teachers is to uncover and nurture this (as well as prodding it along when needed!). As a professor my scholarship looks at democratic practices in mathematics teaching and learning that work toward creating more equitable learning outcomes. See here for some of my academic musings: <a href="http://faculty.fullerton.edu/mellis/Articles%20about%20Mathematics%20Education.htm">http://faculty.fullerton.edu/mellis/Articles%20about%20Mathematics%20Education.htm</a>EllisMathEdhttp://www.blogger.com/profile/04784882038754605959noreply@blogger.com2