tag:blogger.com,1999:blog-14425616014416966972024-03-15T18:09:43.460-07:00The Math SwitchChronicles of inquiry-based learning (IBL) in mathematics and the tools to make learning happen.Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-1442561601441696697.post-81746616904535392382014-10-21T05:38:00.000-07:002014-10-21T05:38:24.601-07:00Teach Like a Freak, Part 2<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
Inspired by the latest installment from Steven Dubner and Steve Levitt, Think Like a Freak, in this series I consider what it might mean to Teach Like a Freak. In part 1, I took up the idea of experimentation and how I am currently experimenting with my teaching. In part 2, I examine two other ideas from Think Like a Freak, targeting small problems and thinking like a child.</div>
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Dubner and Levitt make the point that it often makes sense to target small problems, even if your goal is to solve a large problem. Thinking as a teacher, I think there are times when we can be overwhelmed by the obstacles our students face, both inside and outside the classroom. I think targeting small problems can help a teacher focus and make manageable, lasting changes. As I described in part 1, I am experimenting with regular use of Interactive Engagement questions in my Transition to Proof class. The reason I am doing this is that I felt that it was often difficult to get discussions of student presentations going, and I have been seeking ways to get more lively discussion and broader participation from students.</div>
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Another problem I have targeted is attendance. Students in my classes are absent or late at much higher rates than I would like. Over time, I have tried a number of tactics to solve this problem. I have had maximum allowable absences, which did not work for me, since I did not want to further deduct from students’ grades when (because they missed classes!) they were already in a position where their chances of passing the class were low. Another tactic that I have used with some success is contacting students (via email) when they miss class. Generally, I tell absentees, “We missed you in class,” I may let them know what the next assignment is, and I encourage them to contact me if they wish. My sense is that students get the message that their attendance matters. Of course, I have not experimented (!) to see if I can document the impact of this practice. This semester, I have students submitting responses to IE questions online, and am counting that as part of their grade. Some of the points for those questions are just for submitting a response, so I have effectively made attendance a small part of the grade. I am tracking daily attendance to see if there is an impact. Right now, I still feel like a lot of people are late, but absences seem under control. </div>
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Again stepping back to the larger picture, the main idea of this discussion is to look at teaching not as one monolithic challenge, but as a set of smaller problems, and then to tackle them, either separately or together.</div>
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The authors also present the idea that one should think like a child, meaning that a child is not afraid of wild ideas. A child is not bound by the conventional wisdom. As an example of this idea, there is a current movement called Statway, developed by Carnegie (http://www.carnegiefoundation.org/statway) and the Dana Center, that aims to serve students who would otherwise be in a yearlong sequence of developmental mathematics, and instead give them a semester of developmental mathematics plus an additional course tackling issues not directly about mathematics content (for instance, developing students with the mindset that they can get smarter), and then putting them into a college level statistics course. This certainly seems unconventional on the face of it. The most common response to students struggling in mathematics is to blame their prior knowledge. The Statway approach is to treat students within the larger framework of their approach to learning, and to address those issues. Although I have not seen a lot of data, from what I know, Statway is showing promise.</div>
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In education, especially higher education, we can be victims of our own success. We are the ones who succeeded in education, so it can be especially hard to challenge the norms that, very often, with which we are enculturated. It takes effort to get outside our own perspective, but it can be done. As a recent post from Grant Wiggins (http://grantwiggins.wordpress.com/2014/10/10/a-veteran-teacher-turned-coach-shadows-2-students-for-2-days-a-sobering-lesson-learned/) demonstrates, one way is to shadow a student. If this is not practical, even carrying on a casual conversation with a student outside of class can offer insights into ways we could be better at helping students learn. Statway is an example of finding a way to make a difference by thinking unconventionally. We in academia are proud of our intelligence, innovativeness, and originality, but we need to widen our focus to those areas that have become accepted, and thus, not questioned, if we are to make strides in helping students.</div>
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So, to my fellow educators, get your freak on! Try new ideas, and tackle those small problems.</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com1tag:blogger.com,1999:blog-1442561601441696697.post-32257579179671021792014-10-14T11:10:00.002-07:002014-10-14T11:10:28.193-07:00Teach Like a Freak, Part One<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">Inspired by the latest installment from Steven Dubner and Steve Levitt, Think Like a Freak, in this two-part series I consider what it might mean to Teach Like a Freak. In part 1, I take up the idea of experimentation and how I am currently experimenting with my teaching.</span><br />
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One of the central premises of Think Like a Freak is that one should be willing to experiment, and to make decisions based on the data gathered. For quite a long time, I have been teaching using inquiry-based learning (IBL), a mode of instruction in which students are the focus of classroom activity, deeply engaged in collaboratively making sense of the content. Evidence has been mounting that IBL specifically (<a href="http://www.nctm.org/publications/article.aspx?id=42527">http://www.nctm.org/publications/article.aspx?id=42527</a>) and active learning more generally (<a href="http://www.pnas.org/content/111/23/8410.abstract">http://www.pnas.org/content/111/23/8410.abstract</a>) are more effective than lecture across multiple outcomes. At the same time, I have been reading research about Interactive Engagement (<a href="http://www.ams.org/notices/201308/rnoti-p1018.pdf">http://www.ams.org/notices/201308/rnoti-p1018.pdf</a>), and have been experimenting with trying to blend IE with IBL. The question for me is how to structure my class to take maximum advantage of these approaches. To force myself to take this question seriously, I promised to speak about what I learned at the JMM 2015 in San Antonio.
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Last year, when I taught Transition to Proof, I had made a handful of IE questions, but class time was spent mostly on students presenting their work, and our discussions of those presentations, and a little bit of pair work. What I am doing this semester is using IE questions every week, which means about 15-25 minutes out of 150 minutes are spent on these short questions, with the rest of the time being spent the same way as last year. So, my goal is to compare the two classes in their understanding and skill in writing proof. The next step is to decide how to assess the impact of blending IBL with IE. This involves deciding what to measure. Another issue is that, once I decide on appropriate measures, it can be difficult to get good comparative data. For purposes of assessing impact, I do not have two classes running simultaneously with which to carefully set up a comparison. The best I can do is to use the data I still have from last year’s class. More specifically, in the previous year's course, I had tried a handful of IE questions, and so I have the results from those as well as exam scores for that class. The key component of my assessment, then, is to compare student exam performance last year to the performance this year, when I am using IE questions on a weekly basis. I do not claim that this will give me a definitive answer, but at least it will be a start. Another thing I have been doing is keeping track of participation in whole class discussions, so that I can compare the number of participants in discussions on IE days with discussions on non-IE days. Although counting the number of participants in whole-class discussion is a somewhat superficial measure, it gives me some quantification of how things run differently with the IE questions. If it seems that students are benefiting from more IE questions, I will keep making time for them in class.
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Backing away from the specifics of this question, one thing that I have decided to do with courses that I teach regularly is to keep results of exams broken down by question. The reason for this is that I often modify exams from year to year, and so exam scores from year to year are not directly comparable, but there will be questions that are directly comparable. Another thing that I am learning to do is to keep a log of each class day’s activity. This way, in addition to evidence of student learning, I have a record of the kinds of interactions that occurred in class meetings. Together, these provide two kinds of data that help me to know whether what I am doing is working. Although I have always modified my teaching over time, by Teaching Like a Freak, I can hope to have evidence of whether the changes are making a positive impact.
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In part 2, I will take up two other ideas from Think Like a Freak, targeting small problems and thinking like a child. </div>
Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com4tag:blogger.com,1999:blog-1442561601441696697.post-71677323865407756462014-09-24T07:10:00.000-07:002014-09-24T07:10:25.418-07:006 Responses to Students' Questions About IBL<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">Teaching is a cultural activity. Whenever students enter a classroom, they have expectations about their roles, and about what the teacher will do. If a teacher decides to do something outside of the norm, students are often confused and anxious about what will happen. Students often express these feelings with statements such as, </span><br />
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"Why do we have to teach ourselves?" "I don't learn this way," or, "Why not just show me what to do?"
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Over time, I have accumulated some ways to respond to these statements. I did not develop all of these myself; many of them have come from other instructors I have met at conferences and workshops. I should also say that these are things that I say to college students. If I had an audience of high school students, for instance, I would probably make these same points, but not necessarily in the same way.
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<li>Think of me as a coach. When you think of things you learned outside the classroom, things like playing a musical instrument, learning to swim, or playing a video game, you probably learned a lot of it by trying things yourself. That is what we are doing here. By letting you show me what you tried, I can coach you, and help you figure out anything you are missing. To quote a student of Dana Ernst’s, “Try. Fail. Learn. Win.” In other words, people learn from their own efforts. And, like a swim coach, while I will expect you to try to swim on your own, I will also step in to keep you from drowning.</li>
<li>Learning happens when we are actively involved. A lot of research has accumulated that suggests that classrooms in which students are actively participating and collaborating are better at promoting learning than classes in which students are passive. (Research on cooperative learning for K-12 is summarized in <a href="http://www.amazon.com/Classroom-Instruction-that-Works-Research-Based/dp/0131195034">Marzano, Pickering, and Pollock, 2001</a>, for example. The recent article by <a href="http://www.pnas.org/content/111/23/8410.abstract">Freeman and colleagues</a> reviews research on active learning at the college level.)</li>
<li>Do not mistake struggle for “not learning.” We are often not very good at judging how well we are learning. When people watch a well-organized lecture, they rate their learning higher than when watching a poorly organized lecture. But tests suggest that the audience in each case learns about the same amount. In contrast to a lecture, when we participate actively, it can feel uncomfortable to struggle to come up with answers. But that struggle is part of learning. Consequently, students sometimes rate their learning in lecture courses as higher than in active learning courses, even when the data suggest otherwise. More than one study has pointed to this, but this is apparent in a study by Lake, 2001 (<a href="http://ptjournal.apta.org/content/81/3/896.full">http://ptjournal.apta.org/content/81/3/896.full</a>).</li>
<li>This course will help you develop the skills that employers want, such as independence, creativity, the ability to work in teams, the ability to learn new ideas, and skill in solving problems for which the solution is not immediately apparent. My goals for you in this class are not just to pass exams; I want you to learn skills that will be valuable to your long-term success in your chosen career. But developing these skills requires doing something different than watching the instructor and practicing similar work on your own time. It will require struggle, as you are learning to use a different set of skills than you may be used to using in math class.</li>
<li>A lot of people tell me they hate math, or, “I’m just not a math person.” The kinds of experiences that lead people to make these statements have a lot to do with the way math has been taught for a long time. A lot of classes emphasize following the teacher’s steps, practicing specific procedures, memorizing mathematical facts, and developing speed at execution. While there is a role for these things, mathematics is about a lot of other things, and for the most part, it is these other aspects that interest mathematicians in doing mathematics. The other side of mathematics is about solving problems, finding new ways to understand mathematical ideas, and proving that the solutions we find work, or figuring out the cases where they don’t work. This kind of work does not proceed linearly, from problem directly to solution. Instead, we often take a winding road, hit dead ends, and have to re-evaluate what we are doing. This kind of mathematics is not straightforward, but it is exhilarating when we succeed, and even when we don’t, we often learn a lot.</li>
<li>A good teacher is not defined by what he or she knows, but what he or she can get students to learn. It is what the students can do that matters. I can explain a lot of sophisticated mathematics, but that is no guarantee that you will learn it. Instead, I carefully prepare problems that will help you draw out your own ideas, and that are most likely to put you into a situation where you will learn the important ideas of the class. Then, as a class, we will struggle, but you will learn more than you would if this class was organized around me, as the instructor, explaining solutions to problems you have not yet thought about.</li>
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I’m sure other instructors have other ideas, and I’d be happy to hear them in the comments. Meanwhile, I hope these examples serve to illustrate the kinds of answers that an instructor can use. I find that both the use of analogies (as in #1) and the appeal to research (as in #2 and #3) tend to be my favorite. I probably lean on #1 the most, but I also mention the others regularly. In classes where the students are not STEM majors, #5 often resonates with the personal experience of many students, and can help open the door to having them consider other ways of organizing the classroom that can still be called teaching, or, better still, to think of the classroom as being about what the students learn, rather than what the teacher explains. </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com1tag:blogger.com,1999:blog-1442561601441696697.post-34714877082171232212014-09-16T06:53:00.000-07:002014-09-16T06:53:52.037-07:00Who counts?<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">In this post, I discuss federally defined graduation rates. </span><br />
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If you have not heard, accountability of the sort that has been in place for K-12 for more than a decade seems to be creeping into higher education.
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In some ways, I am happy that this discussion is taking place. As a professor, I take pride in seeing students learn, grow, graduate, and move on to success in their adult lives. I think it is good that institutions of higher education are looking at graduation as a measure of the accomplishments of an institution, and that they are focusing on ways to support students in reaching their goals. However, as often happens, finding an appropriate way to measure graduation is not as simple as it seems, and the current definition does a great disservice to some institutions.<br /><div>
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Much of the discussion of graduation rates is based on the Federal Graduation Rate (FGR), which measures the percentage of <i>first-time, full-time freshmen who graduate within six years of entering their original four-year institution</i>. If you went to college immediately after high school, took classes full-time while working part-time, and graduated within 6 years, this definition probably seems perfectly reasonable on the face of it. If you fit that profile, congratulations. More than half of college students do not.
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For purposes of illustration, suppose we are looking at the graduation rate for “Big State University (BSU).” In the FGR:<br /><div>
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<li>Transfers are not counted for anyone’s benefit. If a student begins his/her college career at another institution (whether a community college or another 4-year institution), and transfers to, and later graduates from, BSU, that student is not counted toward the success of either BSU or the original institution. Neither one!</li>
<li>If a student begins his/her BSU career at less than a full-time unit load (often 12 units), or begins in mid-year, then s/he also does not get counted toward the success (or failure) of the institution. Nothing that happens with that student will impact the FGR reported by BSU.</li>
<li>I have been told, but have not been able to verify, that even if a student withdraws for a semester, or stops taking full-time course loads, that s/he is removed from eligibility to be counted as a graduate.</li>
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Taking these factors into account, the American Council on Education estimates that about <b>61%</b> of students at 4-year schools are excluded from the calculation. <b>Sixty-one</b> percent! How do institutions have a conversation about their successes and shortcomings when they ignore more than half of the students in their calculation?
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To give you another perspective, I looked at my own institution to find out how many of our graduates are being counted toward our success, as measured by the FGR.
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Using public Integrated Postsecondary Education Data System (IPEDS) data from my institution, the data and my calculations indicate that <b>306</b> students who graduated from the university in 2012-2013 were among the students counted in the FGR, meaning that they were first-time, full-time freshmen in one of the years 2007-2010. Does that sound like a cozy and intimate graduation ceremony? I guess that depends on where you went to school. But in fact, the institution awarded <b>2,481</b> bachelor’s degrees in the 2012-2013 academic year. That means that fewer than 1 in every 8 graduates is counted as part of the university’s official graduation rate. How do we define the success of an institution when 7 out of every 8 students receiving degrees, or more than 2,100 of its graduates, do not count toward the accomplishments of the institution?
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The university serves a population with a lot of students eligible for financial aid, and serves a lot of non-traditional students who may work full-time. That is part of the institution’s mission, and one I am proud to serve. But the FGR does not measure our success.
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So, the next time you hear about graduation rates in higher education, be glad that the individuals care about students being able to complete their degree. And then tell them that there is a lot of success that is not being counted.
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I am not the first person to make at least some of the points here. Yesterday, I searched for “federally defined graduation rate” and I got hits that included articles and blog posts making exactly some of the points here. But I don’t think they have looked at the problem from the other end, how many of an institution’s graduates are or are not students who count for the FGR.<br />
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Here is one example of an article discussing some of these issues:</div>
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<a href="http://www.acenet.edu/the-presidency/columns-and-features/Pages/Why-Graduation-Rates-Matter%E2%80%94and-Why-They-Don%E2%80%99t.aspx">http://www.acenet.edu/the-presidency/columns-and-features/Pages/Why-Graduation-Rates-Matter—and-Why-They-Don’t.aspx</a></div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-71773799846336899302014-09-02T06:22:00.000-07:002014-09-02T06:22:05.430-07:00The Excitement of September<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">It’s September, and my wife, my daughters, and I have all started our school years. I am excited, as I am nearly every year at this time. September is the time for hope. I have new students that will be engaged and, I hope, experience the joy of learning. I am teaching a course that I have never taught to a group of students, most of whom I have already had. I am excited about working with new content and seeing the ideas that students will bring to it.</span><br />
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In just two class meetings, there have already been some highlights. In one class, the students were deeply engaged with the concept questions that I used with them. The debate was lively, and this set the tone for good discussions of student presentations. In my other class, we left a question unresolved, but two students turned in two different, but both viable, solutions to the problem, which will lead to a good discussion this coming week.
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I hope your year is off to a good start. </div>
Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-69681573658490746412014-08-19T06:57:00.000-07:002014-08-19T06:57:25.044-07:00What is acceptable evidence?<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">I often try to keep the goals of my course in mind when designing a course, and when making decisions day-to-day in a course. But I have a harder time thinking about evidence of student learning. As it turns out, verbs are helpful.</span><br />
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Through my work with teachers, and in partnering with teacher educators, I came into contact with the book Understanding by Design, by Grant Wiggins and Jay McTighe. In their model for developing curriculum and lessons, there are three stages, embodied in the three questions: What are the learning goals? What is acceptable evidence? What activities, experiences, and lessons will lead to the desired results as evidenced by the assessments? </div>
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I teach a range of courses, some for aspiring elementary teachers, some for math majors, and some for practicing secondary teachers. In planning any of these courses, I generally begin with my learning goals for the course. While the official course syllabus sets a direction for the course, I sometimes find it helpful to rephrase the goals, and to prioritize them. For purposes of illustration, one phrasing of a course goal from the course for future elementary teachers is: Students will understand fractions and their representations, and be able to solve problems involving fractions. My rephrasing of that goal is: Students will be able to <i>explain</i> operations on fractions using models such as the area model and the number line, <i>apply</i> the models in realistic contexts, <i>solve </i>problems involving fractions, and <i>interpret</i> their answers.</div>
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Pivoting from goals to assessment, I am now faced with the question, What am I looking for during and after the unit on fractions that will let me know if students have reached the goals for the unit? Whereas the initial phrasing of “understand fractions” does not translate easily to something that can be assessed, the use of the verbs explain, apply, solve, and interpret give more direction to how to assess student learning. It lets me know that I am going to assess the students via problems in context that include a requirement to interpret their answers, as well as problems that require explanation of diagrams or models. I will know students have succeeded if their explanations are coherent, their diagrams illustrate mathematical reasoning, and students are purposeful in interpreting their answers in context, for instance by using appropriate units, or rounding up or down as appropriate to the problem. Although I am sure it is possible to have shallow goals using the same verbs, I find that using words such as explain, apply, solve, and interpret translates an abstract concept like understanding into a measurable quantity. </div>
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Rethinking my goals with assessment in mind has helped me keep a focus on what is important in my classes. Having clear goals phrased with verbs that make them measurable makes it easier to write exams. But more importantly, because I am interacting with students at every class meeting, if I feel students are not reaching the goals, I know what is important to emphasize, or what is important enough that we need to slow down, since it is embedded in the goal statements.</div>
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Have you worked with your goal statements? Do you find that you assess progress toward your goals regularly? </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-32080462618236941052014-08-12T07:47:00.000-07:002014-08-12T07:47:21.094-07:00Examining Reasons to Use Technology in the Classroom: Mathematical Modeling of Flight Times<!--?xml version="1.0" encoding="UTF-8" standalone="no"?--><span style="font-family: Arial, Helvetica, sans-serif;">
In this post, I explore how technology has made mathematical modeling more accessible.</span><br />
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This summer, I had the privilege of teaching a 3-week institute for eighth grade teachers. One of our aims was to help teachers grapple with mathematics in the Common Core State Standards that is new to (or long forgotten by) the teachers. One of the major changes is the inclusion of a mathematical modeling standard (Standards for Mathematical Practice 4), and in eighth grade, three standards refer to investigating patterns in bivariate data. This includes thinking about whether a pattern fits a linear model, and informally fitting a line to data. Thus, we spent a number of sessions engaged with bivariate data. For purposes of this post, the main point I want to make is how technology has made fairly sophisticated mathematics more accessible, and to briefly describe how we used technology to do mathematical modeling.
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As I have written <a href="http://themathswitch.blogspot.com/2014/07/in-this-post-i-describe-my-experience.html">previously</a>, I think the TI-Nspire is worthwhile in spite of its price, and so that was the focus of much of our work. The problem Gate-to-Gate, which I updated and adapted from a problem I found in a book, is a good example of our work. In brief, the goal of the lesson is to build and assess a model that predicts flight times from Chicago given the flight distance from Chicago.
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In that problem, we started by making observations about a map produced by <a href="http://www.flighttimesmap.com/">http://www.flighttimesmap.com</a> that shows concentric rings labeled with estimated flight times. I used Chicago as the point of origin. First, we collected observations, such as: the rings appear to be circles, the circles appear to be equally spaced, and the first circle is marked as a 1 hour flight. Next, we discussed the meaning of the observations, and we conjectured that the equal spacing is an indicator of a fairly constant flight speed. We also wondered whether the map was completely accurate with its times.
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The next step was to have the teachers explore a data set. I gathered actual flight times that I had looked up and put the data in a TI-Nspire file. The teachers were then saved the trouble of typing in their own data. Teachers then created scatter plots, attempted to fit their own informal lines to the data, and ran a linear regression on the data. With the line, they then chose flight destinations, looked up the flight distances (via web search), and compared the flight times predicted by the model to those they found on the web. They also tried to think of cities on the map that would be 3 hours away from Chicago by air, and again compared the real data to the predictions of the model.
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Finally, we had a summary discussion about the quality of the model fit, the meaning of the values in the linear equation, and considerations about what is an appropriate domain for the linear function. (What does it mean to have a flight covering a distance of 0 miles?) Some teachers graphed distance as the independent variable, and others graphed time as independent, giving two different equations. This led to different insights from the different slope and intercept values. It was a good discussion and led to good insights about both modeling and the meaning of slope and intercept in context.
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Stepping back from the problem, here is a look at how technology enhanced this exploration.
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<li>Data is more easily shared. This saves a tremendous amount of time. I shared the TI-Nspire file as a Dropbox link on a Lino board that I established for the class. This is a long way from having to either plot data points by hand, or even sharing the data but having each person enter data into their own spreadsheet for analysis. If I were not doing this lesson with iPads, I would either have to pre-load data onto handheld calculators, or if those were not available, perhaps give the data in a table and an already-plotted graph (or two graphs, with the two choices of independent variable).</li>
<li>Data is more easily analyzed. Fitting a line informally can be easily explored with touch screens. And, since the technology handles finding the equation of the moveable line, the focus of the conversation is on the quality of fit of the model, rather than a focus on the procedure of finding the line equation. Computing technology to perform tasks such as regression has been available for many years. Nonetheless, it is a powerful tool, and the ability to use regression with a button click means that there can be a discussion of how our informal lines compared with the regression line. If this were a calculator lesson, we might still use moveable lines, but less easily. And, barring that, we would have spaghetti on a paper graph, but then we would not be able to compute the line equations quickly.</li>
<li>Access to the web helps make it easier to test a model with real-world data. With access to maps and the ability to look up flights, teachers had a lot of freedom to test their models. If we did not have the web, I would have had to preselect a set of cities, listed with distances and flight times, and use that as the basis for testing the model.</li>
<li>Sharing results is easier. We used Baiboard, and I selected individual teachers, who then uploaded screen shots of their models and results. This meant that when teachers were sharing, either they or I could add annotations to the screen shots. Moreover, as others shared, we could swap back and forth between the current person’s work and the work already shared by others. If this were a lesson on calculators, teachers would have had to keep a separate handwritten record of their work, and switch back and forth between sharing their written work and sharing the work on the calculator. We would probably have to keep a (partial) record of what was shared on a whiteboard for later reference.</li>
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In looking at the effect of technology on the lesson, it is not the case that without iPad technology, the lesson is impossible. Compared with, say, having classroom calculators, it is that the technology makes the lesson run more smoothly and quickly, adds the authenticity of finding one’s own data, and improves the way results can be shared.
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com1tag:blogger.com,1999:blog-1442561601441696697.post-85483327059280419622014-08-05T06:24:00.000-07:002014-08-05T06:24:26.643-07:00Keep Tinkering<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">I am always making adjustments, tinkering with my courses, both during but especially between iterations of the courses. My teaching is never a finished product. It is in the nature of teaching that what worked in one year for one course may not work for another course or in a subsequent year of the same course. I want to share one change that I have made over the past year, the effect it had, and what I am doing as a result.</span><br />
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In my Transition to Proof course last fall, I began building concept questions to supplement the regular proofs, and using them to target specific misconceptions or difficulties that students are having (or that I expect based on past experience). By concept questions, I mean short questions, usually multiple choice or true/false, that are designed to draw out students’ thinking and generate productive disagreement. Every time we had one of those discussions, I was exhilarated by the amount of discourse in the room. This practice evolved because I promised myself that I would focus on getting more discussion out of students in that class, since, in the past, I felt that there were too few students able to comment or question the proofs presented by their peers at the board. With the concept questions, I felt like I was seeing what the students were getting or missing from those proof presentations. In particular, the questions really helped to draw out the main points of proofs, points that I thought they would have gotten from a direct discussion of the proof, but which may have been less apparent than I had assumed. I almost feel like students in previous iterations of the course were shortchanged because they did not get this added layer of discussion to push their thinking forward. That’s when tinkering pays off.</div>
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As a result, I have planned some form of concept questions into both of my courses for this fall. Accompanying this change, I have also included the concept questions into the course grade. In addition to using the concept questions as a teaching tool, I am curious as to: (a) whether simply participating in the concept questions correlates with performance in the course, and (b) whether answering questions correctly on the first try correlates with performance in the course. Most of all, I would like to know whether using the concept questions as a tool in class improves the class’ understanding of the key concepts, but this will be hard to measure. I am thinking that I may have some items on some exams in one of the classes that I will reuse from prior years, so that I can compare performance. That’s not as good as an experiment, but at least I will have a basis for comparison.</div>
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The larger message is that it is healthy to revisit one’s goals for a course, to think about personal goals for improving one’s teaching, and to be willing to try new ideas that show promise of bringing students closer to the learning goals, and to measure the impact of the changes so that what works remains in place, and tactics that don't work are revised or edited out of the course. </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com1tag:blogger.com,1999:blog-1442561601441696697.post-55018935320163678352014-07-25T05:54:00.000-07:002014-07-25T05:54:06.812-07:00Most popular posts from one year of blogging<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
The Math Switch began one year ago, in July, 2013. In that time, I have enjoyed sharing ideas on inquiry-based learning and on educational technology. In the last few months, I have been able to post regularly, at 3 Tuesdays per month. Over that time, the most popular posts have been:
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<li><a href="http://themathswitch.blogspot.com/2013/08/9-ways-to-engage-reluctant-students-aka.html">9 Ways to Engage Reluctant Students, aka Tackling the Startup Problem</a> </li>
<li><a href="http://themathswitch.blogspot.com/2013/08/harnessing-your-personality.html">Harnessing Your Personality</a> </li>
<li><a href="http://themathswitch.blogspot.com/2013/10/dealing-with-misconceptions-part-1.html">Dealing with misconceptions, Part 1: Seven ways to handle misconceptions in the moment </a></li>
<li><a href="http://themathswitch.blogspot.com/2013/11/engage.html">Engage!</a> </li>
<li><a href="http://themathswitch.blogspot.com/2013/09/a-critical-examination-of-my-transition.html">A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins </a></li>
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Thank you to all who have stopped by to read the posts. A special thanks goes to those that have re-shared, or commented on the blog. </div>
Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-84986070387129849672014-07-15T06:39:00.000-07:002014-07-15T06:39:24.941-07:00An Introduction to 6 Apps for Quizzes and Polls<!--?xml version="1.0" encoding="UTF-8" standalone="no"?--><span style="font-family: Arial, Helvetica, sans-serif;">
In this post, I discuss 6 apps and websites for quizzes and classroom polls. This is not a deep look, but I will tackle some critical basic features: the types of questions available, the kinds of resources that can be embedded in the questions, and what students or participants need in order to respond. All of these 6 apps are free, at least up to a certain usage level.</span><br />
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<span style="font-family: Arial;">Readers may also wish to consult the comparison chart at </span><a href="http://www.polleverywhere.com/vs" style="font-family: Arial;">http://www.polleverywhere.com/vs</a><span style="font-family: Arial;"> (the chart dates from November, 2012), and to look at some of the information provided by Richard Byrne at </span><a href="http://www.freetech4teachers.com/2013/03/four-good-alternatives-to-clicker.html" style="font-family: Arial;">http://www.freetech4teachers.com/2013/03/four-good-alternatives-to-clicker.html</a><span style="font-family: Arial;"> and elsewhere on his site.</span><br style="font-family: Arial;" /><ol style="font-family: Arial;">
<li><b>Edmodo</b>: <a href="https://www.edmodo.com/">https://www.edmodo.com</a> is a course management system, with quizzes and polls as embedded tools. Edmodo has multiple choice, true/false, short answer, fill in the blank, and matching quizzes, as well as multiple choice polls. Quizzes have a number of nice features, including the ability to embed links, video, images, and LaTeX (by enclosing the mathematics with [math]…[/math]). Polling is simpler, with just the multiple choice mode and no embedding. Students should have accounts and be set up in a class in order to use either polls or quizzes.</li>
<li><b>Socrative</b>: <a href="http://socrative.com/">http://socrative.com</a> has multiple choice, true/false, and short answer formats for both polls and quizzes. Quizzes can have embedded images, but not video or links or LaTeX (unless you create an image with LaTeX in it). Quizzes can be run as a game called Space Race, where getting answers right moves a rocket across the screen in a race with other participants. In a quiz, students can get immediate feedback on whether their answer was correct if the quiz is set up with the correct choices marked. Alternatively, if the correct choices are not marked, students do not immediately know if they responded correctly. Polls (“Single Question Activities”) can be run instantly, with no need to pre-load questions. In a poll, the idea is to set everything up <i>without</i> Socrative, and just use Socrative to collect votes of A/B/C/D/E, where the instructor can designate what each response means. Student accounts are not needed. Just recently, accounts have been switched over to Socrative 2.0. Socrative 2.0 adds a feature, Exit Ticket, which is pre-formatted with three questions: a multiple choice question about how well the student feels he/she has learned the day’s lesson, and two short answer responses, one a request to describe what was learned, and the second to answer the teacher’s question (which allows the teacher to pose a specific question, i.e., outside the app, in addition to the general one). </li>
<li><b>Google Forms</b> are part of the suite of Google Drive tools. Forms support multiple choice, multiple correct, short answer, and fill in the blank. Forms can have embedded images, video, or links, but not LaTeX (unless you create an image with LaTeX in it, as I described <a href="http://themathswitch.blogspot.com/2014/06/typesetting-mathematics-in-google-drive.html">earlier</a>). Students do not get immediate feedback about the correctness of their answer choices. Auto-grading of the responses can be accomplished by installing the Flubaroo script in Sheets. Students do not need accounts. However, to get the maximum benefit from Flubaroo, it is a good idea to collect student emails in the Form.</li>
<li><b>Quiz Bean</b> is web-based, and not an app. It has multiple choice, true/false, and multiple correct formats. Quiz Bean supports embedded images, but not video or links or LaTeX (unless you create an image with LaTeX in it). Students get immediate scoring feedback as they progress through the quiz. Students need accounts and accounts should be set up into a class by the instructor.</li>
<li><b>Quizlet</b> is built more as a study tool. After setting up an account, users build virtual index cards and then practice quizzing themselves, matching the items in one of a few ways. The index cards can include images or text. </li>
<li><b>gFlash+</b> is similar to Quizlet in that it is designed for building virtual index cards. The “g” indicates that the index cards can be created from Google Sheets. There is no need for a gFlash+ account, but this app works best if connected to a Google Drive account.</li>
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Besides the ones listed above, there are many, many more. An incomplete list of them includes:
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<li>Exit Ticket: <a href="http://exitticket.org/">http://exitticket.org</a></li>
<li>Kahoot: <a href="https://getkahoot.com/">https://getkahoot.com</a></li>
<li>Mentimeter: <a href="https://www.mentimeter.com/">https://www.mentimeter.com</a></li>
<li>ParticiPoll: <a href="http://www.participoll.com/">http://www.participoll.com</a></li>
<li>Poll Everywhere: <a href="http://www.polleverywhere.com/">http://www.polleverywhere.com</a>. </li>
<li>TAPit: <a href="http://theanswerpad.com/">http://theanswerpad.com</a></li>
<li>Flisti: <a href="http://flisti.com/">http://flisti.com</a></li>
<li>Infuse Learning: <a href="http://www.infuselearning.com/">http://www.infuselearning.com</a></li>
<li>Quiz Socket: <a href="http://www.quizsocket.com/">http://www.quizsocket.com</a></li>
<li>Geddit: <a href="http://letsgeddit.com/">http://letsgeddit.com</a></li>
<li>Top Hat: <a href="https://tophat.com/">https://tophat.com</a></li>
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<span style="font-family: Arial;">I hope this spurs some ideas. There are so many ways to collect feedback from students!</span></div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-7670866798516026282014-07-08T06:27:00.000-07:002014-07-08T06:27:41.244-07:00Doing math on iOS<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">In this post, I describe my experience using various apps to do mathematical computations. This is focused on the kind of mathematics that arises in K-14 classes, and not research-level work.</span><br />
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Here is the list of iOS apps I have tried for doing math of various sorts on my iPad:
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<li>TI-Nspire CAS is the most valuable app for the iPad. Although it is pricey at $29.99, it is designed for extended exploration in a way that most other apps are not. This has been my go-to app in my work doing mathematical modeling (e.g., linear regression) with middle school teachers. Some of my favorite features include the ability to graph multiple functions or multiple regressions on the same graph and the ability to export files to Dropbox or elsewhere. The export feature allows me to input data to a spreadsheet and share it, thereby saving everyone else from entering data (and making typos).</li>
<li>Wolfram Alpha is versatile, as long as one is interested in looking at one object at a time. By this I mean that one can easily graph any function or set of functions, plot a data set and perform regression, or do standard calculations, but it is not possible to store the results within the app. Instead, it is necessary to take screenshots or copy-paste information to another location (Evernote, for example). The app also makes it difficult to edit information because it is not possible to scroll through a long command line that has been entered. On the other hand, if given an equation, it can show the steps involved in solving the equation. The app can also serve as a search tool to answer questions or provide information. The app requires an active internet connection at all times.</li>
<li>MyScript Calculator is a lot of fun for basic calculations. It transforms hand-written mathematics into typed math script and performs the calculations indicated. It should be noted that getting formatting correct is sometimes difficult, say if there is a rational expression with exponents in the denominator, but it works well for quick scratch calculations.</li>
<li>Geogebra is a spectacular app for the desktop or laptop, but the iOS app has a long way to catch up. What is missing are the settings. For instance, I have never found a way to use a non-square scaling, such as I might need for an exponential function, where the outputs grow much faster than the inputs. Neither does there seem to be a way to adjust the labels (e.g., to show the label <b>on</b> a function), or to display a table of values. Unlike the Nspire or Wolfram, Geogebra does not render 3-dimensional graphs. Still, the app is free, and is good for a lot of Euclidean geometry and 2-dimensional graphing, and it offers sliders for dynamic exploration as well.</li>
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The following are apps that I have used, but not extensively:
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<li>Geometry Pad uses the freemium model. I have used only the free version, which includes the ability to draw basic geometric objects. The premium version adds a lot of features, including the ability to do calculations, graph functions, and a lot more.</li>
<li>Sketch2Graph takes a hand-drawn graph, converts it to a plot of a linear or quadratic function or conic section, and outputs the equation describing the plot. The function graph can then be manipulated by hand. This enables some nice exploration of these graphs and the relation between the graph and the equation.</li>
<li>Algebra Tiles is designed for illustrating or manipulating algebra tiles in an app. The interface has three modes, basic, equations, and factors. This app works as a tool, and is not built to give practice problems nor does it show how to use the tiles. It does serve as a functional replacement for using actual tiles.</li>
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Readers, what have I missed?</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com2tag:blogger.com,1999:blog-1442561601441696697.post-67057520228950863632014-07-01T03:49:00.000-07:002014-07-01T03:49:25.486-07:00Taking notes on iPad: Evernote, NotesPlus, and Notability<span style="font-family: Arial, Helvetica, sans-serif;">In this post, </span><span style="font-family: Arial;">I briefly describe situations in which I find I need to take notes on my iPad, and the apps that I find most useful in these situations: Evernote, NotesPlus, and Notability.</span><br />
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There are two recurring situations in which I find I need to take notes with my iPad. The first is in meetings. In these situations, usually I am able to type notes as the meeting is happening. In this case, I use <a href="https://itunes.apple.com/us/app/evernote/id281796108?mt=8">Evernote</a>. Evernote is perfect for typing notes because I can open the app and start a note almost immediately. Occasionally, there may be a one-sheet handout as well. (Generally, if the handout is longer than one page, the presenter will share it via email.) I have scanned a number of handouts and the scans have been clear. In addition, items scanned into Evernote become searchable. Occasionally, someone hands out a business card, in which case I scan that in too. For certain recurring meetings, I have a particular notebook where I keep all my meeting notes, or I have a tag for the committee that I use to make sure I can find the note later.</div>
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The second situation in which I take notes is during class. In this case, I find I prefer to handwrite my notes, rather than typing, because I may need to write mathematics. Sometimes I am writing something that I want to share with the class, and other times I am making notes about what is happening during presentations or group work that I want to remember for later discussions or follow-up. This includes the possibility that I photograph student work and then annotate it. I use <a href="https://itunes.apple.com/us/app/notes-plus/id374211477?mt=8" style="font-family: Arial, Helvetica, sans-serif;">Notes Plus</a><span style="font-family: Arial, Helvetica, sans-serif;"> or <a href="https://itunes.apple.com/us/app/notability/id360593530?mt=8">Notability</a> </span>for class notes. I can recommend both apps. For NotesPlus and Notability:</div>
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<li>Both can be backed up to Dropbox. </li>
<li>Both offer an eraser as well as an undo button. </li>
<li>Both offer a close-up box for writing. </li>
<li>Both apps have always retained everything I’ve created. I have never experienced disappearing notebooks or pages.</li>
<li>Both apps offer a variety of pen thicknesses and colors as well as a highlighter.</li>
<li>Both apps offer the ability to add audio recordings to notes.</li>
<li>Line segments are handled differently. In NotesPlus, drawing line segments is integrated into the note. For instance, if I want to draw a rectangle, I begin drawing it, and it appears where I put it. Usually, NotesPlus auto-detects the line segments and gives me control points to adjust the placement of the segment. In contrast, in Notability, when drawing segments, the app takes me out of the space where I am working, and then I have to insert the line drawing back onto the page. When I insert the drawing, there is white space around it, so it feels like inserting a picture into a document. </li>
<li>Typed notes are handled differently within each app. NotesPlus offers text boxes that can be inserted anywhere, whereas in Notability the options are to insert stickies or to move the cursor around on the page. </li>
<li>NotesPlus has a built-in web browser, in case one is looking to clip information from websites to insert into notes, a feature not present in Notability. </li>
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Together, this set of tools has really helped me get the most productivity from my iPad.</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-5230028895502064472014-06-17T05:30:00.000-07:002014-06-17T06:09:07.821-07:009 Books to Read and Reread<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial;">In this post, I offer some suggested readings that I find help inform my approach to teaching. The books are listed in no particular order. </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOE8wDC7sEHTBY3EDXLkq3EX2WTPDpuoDZZjPhgeHIvOoFii2Yo0clcKzBH3yTxD8w0vd-rRad5p9c9QK6-c_ZHPnbc9_L4Ucbw2lVn1fZ2gW1VF-K9Ti_UB9IVitvYEp3cKL2c0W6Bx09/s1600/IMG_0980.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOE8wDC7sEHTBY3EDXLkq3EX2WTPDpuoDZZjPhgeHIvOoFii2Yo0clcKzBH3yTxD8w0vd-rRad5p9c9QK6-c_ZHPnbc9_L4Ucbw2lVn1fZ2gW1VF-K9Ti_UB9IVitvYEp3cKL2c0W6Bx09/s1600/IMG_0980.jpg" height="239" width="320" /></a></div>
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For any teacher, I recommend:
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<li><i><b>What Works in Schools?</b></i> Robert J. Marzano, Debra J. Pickering, and Jane E. Pollock. (Note that there is now a <a href="http://www.ascd.org/publications/books/102271.aspx">second edition</a> available with a substantially different organization. Either edition is valuable.) This book (first edition) discusses nine strategies shown by research to be effective in improving student learning outcomes. Sometimes the strategies are “obvious,” but it can still be helpful to be reminded that they are important teaching tools. For instance, summarizing and note-taking are effective. However, for me, many of my students have never been taught how to take notes, or have never discussed strategies for taking notes. So I make an effort to tell students when someone states an idea that I think everyone should write down, and I set aside some time for students to discuss what they should write down during class. Other strategies take a more concerted and planned effort to implement. For instance, generating and testing hypotheses is another strategy. While this is a natural part of doing mathematics, this reminds me to include tasks in which students do more investigative work. More than a list of nine ideas, the book has specific recommendations that are helpful. For instance, what are some important features to make cooperative learning successful? These are the kinds of specifics that are discussed in the book.</li>
<li><i><b><a href="http://www.danielwillingham.com/books.html">Why Don’t Students Like School?</a></b></i> Daniel Willingham. Willingham is a cognitive psychologist who poses some key questions and answers them from the perspective of his discipline. There are a few things that I like about this book, and that make me go back to it. One of the things I like is that each chapter closes with implications for the classroom. For example, one chapter discusses our human tendency to prefer and make sense of things as stories. In a course like precalculus, this might be used to frame “telling the story of a function,” where a function has properties like limits as x goes to infinity, asymptotes, periodic behavior (or not), symmetry, and so on. In calculus or analysis, the story idea might be put in terms of the central “conflict,” will a sequence converge or not, or another, is a function continuous or not. Rereading (or skimming) this book and thinking about the implications often inspires me to find ways to improve my day-to-day plans.</li>
<li><a href="http://www.amazon.com/Whats-Point-School-Rediscovering-Education/dp/1851686037"><i><b>What’s the Point of School?</b></i> </a>Guy Claxton. Claxton describes what he believes are the core goals of an education. These are big-picture concepts like developing people who are curious and are lifelong learners. While this is not a book that I return to for help in thinking through the details of teaching, I find that it helps to remind me of what is really important in my role as an educator.</li>
<li><b><a href="http://heathbrothers.com/books/switch/"><i>Switch</i>.</a></b> Chip and Dan Heath. This book inspired the name of my blog. The Heaths describe how to make a switch—a change—either in yourself or others. The single most important idea is that a lot of what we do is driven by emotion, and so we need to think in those terms when looking to effect change. The authors go through several ways of activating the emotions that will enable a switch to happen. I have returned to the book many times, for example, to remind me of how to approach students who are struggling, to help them find the emotion that will drive them to turn around their performance in my classes.</li>
<li><b><a href="http://www.authenticeducation.org/products/books.lasso" target="_blank"><i>Understanding By Design</i>.</a></b> Grant Wiggins and Jay McTighe. This is a book that puts forth a framework for thinking about curriculum design by starting with the end results, then thinking about how those results will be measured, and only then moving into designing the learning activities that will produce the desired end results. I return to this book from time to time to remind myself of how to frame my goals, and how to find ways to measure progress towards those goals.</li>
<li><b><a href="http://mindsetonline.com/"><i>Mindset</i>.</a></b> Carol Dweck. Dweck has done significant research into the power of having a growth mindset, a mindset in which one believes that through hard work, one can get smarter or better. In the book, she describes some of this research and how it can make a difference across different domains of school and life. The book helps to remind me of why a growth mindset matters, and serves up examples that I use in explaining the power of the growth mindset to students.</li>
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For math teachers at any level, I recommend:
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<i><b><a href="http://joboaler.com/" target="_blank">What’s Math Got To Do With It?</a></b></i> Jo Boaler. Boaler has studied high school students experiencing problem-based curricula and compared them with those in traditional curricula in two different countries, the US and the UK. This book describes some of what was learned in those settings, and distills for a general audience—including parents of schoolchildren—some of the key ideas of what mathematics learning is, or should be, about. From the perspective of a math teacher, this book is less likely to offer ideas for day-to-day decisions, but like Claxton’s book, helps to remind me of the goals of teaching mathematics.
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For college teachers, I recommend:
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<b><a href="http://www.bestteachersinstitute.org/id51.html" target="_blank"><i>What the Best College Teachers Do</i>.</a></b> Ken Bain. Bain’s book centers how the select group of highly-respected teachers he studied approach teaching, from preparing for class, to setting expectations for students, to conducting class, and so forth. Each chapter holds a wealth of good advice, like seeking the commitment of the students: asking them to consider whether they are willing to do what it takes to succeed in the class, and therefore have them commit to the effort required. I find I sometimes return to the questions he poses in the chapters as a way of gaining a fresh perspective on my courses.
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Finally, for college math teachers, I recommend:
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<b><a href="http://www.maa.org/publications/ebooks/the-moore-method" target="_blank"><i>The Moore Method: A Pathway To Learner-Centered Instruction</i>.</a></b> Charles A. Coppin, W. Ted Mahavier, E. Lee May, and G. Edgar Parker. The four authors of this text each describe how to implement the Moore method, as they see it. The book offers the reader a chance to consider various aspects of teaching in a learner-centered environment, and benefits from the approach of the authors, which is essentially to offer their individual responses to the key questions in setting up and operating a Moore Method course. This variations-on-a-theme approach has the effect of providing the reader with a canvas and a palette, rather than promoting a specific paint-by-number prescription. The authors take on a wide variety of issues associated with implementing the Moore method, including such topics as, What if no one has anything to present? How do I grade? and many others. I have returned to the book many times to seek out new ideas of how to handle syllabus construction, or to remind myself of ways to approach managing an IBL classroom. (In full disclosure, I should mention that I am personally acquainted with the authors, and have worked closely with Ed Parker.)</div>
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What are some of your favorite or most inspiring reads from the educational realm?</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com4tag:blogger.com,1999:blog-1442561601441696697.post-82570343952025351502014-06-10T06:26:00.000-07:002014-06-10T06:26:49.035-07:00Typesetting Mathematics in Google Drive<!--?xml version="1.0" encoding="UTF-8" standalone="no"?--><span style="font-family: Arial, Helvetica, sans-serif;">
In a <a href="http://themathswitch.blogspot.com/2014/04/working-in-cloud-part-2-google-drive.html" target="_blank">previous post</a>, I mentioned that in the native Google formats on Google Drive, it is difficult to typeset mathematics. Here I will describe how to work around this issue. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Options for creating images with mathematical symbols are:<br />
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<span style="font-family: Arial, Helvetica, sans-serif;">In general, the first ways that come to mind to typeset mathematics are via LaTeX, MathJAX, by using Equation Editor for Word, or by using Math Type. </span><span style="font-family: Arial, Helvetica, sans-serif;">However, in order to adapt any of these to a Google Doc or Form, an image (JPEG or PNG) is needed. Since I prefer LaTeX, I use a LaTeX to image compiler to create an image that can be inserted into a Form or Doc. I have been using <a href="http://www.sciweavers.org/free-online-latex-equation-editor" target="_blank">SciWeavers</a>. The site is not perfect, as it can be difficult to get justification and alignment right, but it does the job. You can copy-paste the image or copy the URL and let the Google Form retrieve the image that way. For a Form, it makes the most sense to typeset your entire question and copy it.</span></div>
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<li><span style="font-family: Arial, Helvetica, sans-serif;">Typeset mathematics in your favorite way. Then take a screenshot and crop it so that it has only the math that you want. Finally, copy-paste the image into the Google Doc, Form, or whatever you are working on.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;">Create the entire document outside of Google Drive, but load it into Drive for the purposes of sharing or whatever other reason you want the item in your cloud account.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><a href="https://chrome.google.com/webstore/detail/gmath/hhaencnpmaacoojogjkobikbmkhikjmm?hl=en-US" target="_blank">g(Math)</a> is a Docs add-on that renders math formulas and graphs in Docs. I have not tried it, but this at least solves the problem for Docs (but not Forms). </span></li>
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<span style="font-family: Arial, Helvetica, sans-serif;">These are either incomplete or inelegant solutions. I am hoping that by posting, either someone will suggest a better alternative, or that Google will address the issue.
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<span style="font-family: Arial, Helvetica, sans-serif;">By the way, for those of you using Forms with your math classes, I am working on a list of some alternatives to Forms that are out there—and there are a LOT of options. This will be the subject of a future post.
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<span style="font-family: Arial, Helvetica, sans-serif;">After I composed this post, I wanted to make sure everything was up-to-date, and I found this thread <a href="https://productforums.google.com/forum/#!topic/docs/mMQl4IkKG2c">https://productforums.google.com/forum/#!topic/docs/mMQl4IkKG2c</a> that includes the same suggestions for Forms.
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-34475645309081913352014-06-03T06:11:00.000-07:002014-06-03T06:11:19.744-07:00Joys of Teaching<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
In this post, I explore the question: What are the joys of teaching?
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<span style="font-family: Arial;">Teaching is something I have been doing for a long time, going all the way back to when I was a student and a tutor. Early in college, I decided that I wanted to teach at the college level. (Immediately before that I had planned on a career in engineering.) If I had to put my finger on what made me want to teach then, I think it was that, as a tutor, I had seen the joy of connecting people to mathematics, a discipline that is associated with a lot of negative experiences for many students. But, as with so many things in life, as we grow older and gain more experience, what we appreciate changes. Nonetheless, seeing students light up when they grasp an idea remains a particular joy for me.</span><br />
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I enjoy teaching because I enjoy the challenge. Each semester, I find myself faced with challenges: I am challenged to react to students’ misconceptions that I have not seen before, or I am challenged to get students to invest their best effort, or I am challenged to find new ways to keep students engaged in class. Every year, I reflect back on what has happened in my classes, I review the evidence of student learning, and I think over the learning experiences that students had. I always see a need to do better. I always rethink my course problem sets and grading scheme, and I look for ways to remake them in ways that will encourage students to learn more from the course.
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I enjoy teaching because I enjoy connecting with students. More than just teaching content, teaching is a coaching and mentoring relationship. I have been at my school long enough that I have had a number of students at multiple points in their careers, sometimes across lower and upper division courses, or in their undergraduate major and in master’s courses or teacher professional development institutes. Sometimes, students complete a course with me and continue to come back for advice or assistance. I enjoy seeing the students grow and gain new perspectives on what they have learned, or seeing them begin to transition from thinking about the classroom from the student’s view to thinking of themselves as teachers.
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I enjoy seeing the impact of a positive learning experience for students in a way that I cannot see through the other things that I do. Being in front of, in the middle of, and generally in the presence of students gives me opportunities to impact students in ways that are not visible when I am in the role of researcher, or serving the university.<br /><div>
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This is a time of year for reflection, and for planning. I encourage my fellow instructors to take a moment to enjoy the fact that teaching is an awesome responsibility, and a great privilege. Enjoy teaching in spite of, and perhaps because of, the challenges!
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Feel free to share your favorite joys of teaching in the comments. </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-69848099952421208382014-05-20T07:10:00.000-07:002014-05-20T07:10:09.264-07:00 Building an Effective In-class Learning Environment, Part 3: Balancing Group Work And Student Presentations<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
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In this series, I explore the questions: What are some advantages and disadvantages of group work and student presentations? How can students be held accountable for learning in groups and from student presenters? What defines a good balance of group time with whole class presentations? In <a href="http://themathswitch.blogspot.com/2014/05/building-effective-in-class-learning.html" target="_blank">Part 1</a>, I focused exclusively on group work.</div>
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In <a href="http://themathswitch.blogspot.com/2014/05/building-effective-in-class-learning_13.html" target="_blank">Part 2</a>, I focused mainly on student presentations. Finally, in Part 3, I discuss considerations involved in balancing time allocated to each of these modes of classroom organization, and balancing the strengths and weaknesses of the two modes against each other. </div>
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<b>Striking a balance between group work and student presentations:</b></div>
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In Part 1, we learned that group work is an effective way to organize classroom learning, but that there are sometimes issues that need to be resolved or discussed by the entire class, or problems that many groups are unable to resolve on their own. In Part 2, we learned that student presentations are good for putting the focus of the class on a particular solution, but that there is the potential for students’ preparation for class and participation in discussion to suffer. Therefore, I find that using both groups and individual presentations helps to keep students engaged in class, and that a good mixture will encourage students to prepare for class on their own time.</div>
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Given the challenges and opportunities associated with group work and student presentations, what mix of these two forms of classroom organization is best?</div>
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For me, there is no one right answer. Even for a particular semester with a particular section of a class, I am sure that different blends of class organization would be valuable. Still, I have found that I tend to favor more groups or more presentations in different sorts of classes. I use two basic models in my classes. In what follows, I will describe the models and why I feel that each one is valuable in the particular courses where I use it.</div>
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<b>Group-centered model:</b></div>
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In this model, roughly 60-70% of each class period is spent in groups. Usually, I have classes that meet twice each week. Typically, on one of these two days, class begins with me introducing the topic of the day, in some courses explaining the manipulatives we will be using, or the calculator functions that they may need to do the day’s mathematics, and then sending the groups to work on problems. I may tell the groups to give a report when they have a solution to a particular problem. As groups work, I monitor their progress, check to ensure that everyone is participating in the work of the group, and ask questions as needed to help groups make progress in their thinking. If a group is ready to report, then I ensure that all group members contribute to the report, and if they answer all my questions satisfactorily, then I approve the report; otherwise, they are told to work more and call me back when they are ready. If I did not request group reports, then I am usually making note of which groups have done work that I think should be discussed in front of the class, and which problems are sticking points for groups. If all groups become stuck, then we transition to a student presentation or a whole-class discussion of how to proceed. Otherwise, groups continue to work until I feel that most of the class is ready to discuss the key ideas and the work that I have identified for presentation. Presentations then serve as a way to codify the important concepts, as a way to compare different solution ideas, and for groups to ask questions regarding issues they had while working. Everyone is sent home to work on problems, and to come back ready to discuss solutions.</div>
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When students return for the next class, they begin in groups right away. Sometimes, I will announce a jigsaw, so that particular groups are assigned to focus on solutions to a single problem and prepare the explanation they will give later. Other times, I make sure that everyone has a colored pen, and I quickly identify which problems will be presented and who will present them, so that we move into student presentations rather quickly. Because students are using colored pens, I can tell what they have done on their own time, and yet taking good notes on the presentations can boost their homework score. It may happen that after a particular presentation, students have the tools they need to solve other problems on which they were stuck, in which case they get time to work in groups again. Or, I may have follow-up problems that build on what was presented, and again the groups are charged to apply what they have learned from the presentations. Depending on time, we may begin a new cycle of looking at a new topic while working in groups.</div>
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I have used and refined this model since I first had my own classes. I find that this is a good model for lower-division mathematics, including courses like Mathematics for Elementary Teachers, where a number of the problems involve computations and generally involve more familiar or concrete concepts. The problems lend themselves to groups being steadily engaged. It is more difficult to use this model when the problems are longer and more abstract. One reason for this is that the average time to solve a problem is longer. This makes it more difficult to launch into a topic during class time and have sufficient progress made by all groups within 30 minutes or so. Therefore, in classes like Transition to Proof, Abstract Algebra, or Modern Geometry, I use a different model.</div>
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<b>Presentation-centered model:</b></div>
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In the presentation-centered model, roughly 70-80% of the time is spent on student presentations (this includes the think-pair-share time in which partners are discussing presentations). Class begins in one of three ways. Either, a) students are encouraged to discuss their solutions while I ensure that everyone has a colored pen and I sign up the presenters for the day; b) the class begins with a set of prompts, in which I put up a short set of questions, often true/false or multiple choice, and students are asked to think-vote-discuss-revote, similar to Interactive Engagement <a href="http://www.physics.indiana.edu/~sdi/ajpv3i.pdf" target="_blank">in physics</a> and elsewhere; or, c) I announce a jigsaw, and partners are assigned to one of two problems that they will shortly have to explain to another person. At the conclusion of any of these events, we launch into student presentations. Each presentation is discussed in detail, until the class is satisfied with the mathematics, and I am satisfied that the class has identified the important ideas. Occasionally, in between presentations, partners may be asked to look at a related problem that either applies the ideas from the most recent presentation, or anticipates the ideas that may come up in the next presentation. After the conclusion of all the day’s presentations, usually four to six of them, then I may point students to the next topic or assignment, and partners will often be asked to do some preliminary work with <a href="http://themathswitch.blogspot.com/search/label/%23definitions" target="_blank">definitions</a> or examples that may help them.</div>
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When I first began teaching proof-oriented courses, I used presentations and accompanied them with think-pair-share, as I do now, but I did not use the jigsaw and prompts. I find that beginning the class with the partner work gets the class into a discussion-oriented mindset, which helps to make the presentation discussions more lively. Using the prompts makes for a nice formative assessment where I learn where the whole class stands with key concepts, and I can see and react immediately to what the class thinks. I also find that students are very highly engaged during jigsaws, so that I often structure the problem sets so that there are two closely related, more accessible problems that lend themselves to a jigsaw. But because a jigsaw depends on a large portion of the students being able to solve the assigned problems, not everything can be handled this way.</div>
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<b>Final comments:</b></div>
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Stepping outside of my own classroom, I know that different instructors have preferences for whole class or small group mode. Each mode demands slightly different skills from the instructor. In small groups, the instructor has to travel from group to group, listening and occasionally contributing questions, and making mental or written notes about the discussions for later summative activities (whole-class presentations or sharing, or instructor summary). The noise and activity level tend to be high. With whole class presentations, the challenge is to ensure that all students are engaging with the content of the presentation, and to do as much as possible to have broad participation. Ultimately, the goal is to have as many students as possible engaged in creating mathematics and making sense of the core ideas of the course for themselves, so that students develop the mathematical thinking skills that will serve them long after the course is over. One of the benefits of inquiry-based learning and the active modes of instruction described here is that there are many opportunities to gain evidence of students’ thinking—to conduct formative assessment, so that adjustments to instruction can be made before an exam reveals critical gaps or misconceptions among the students. And, as the recent Proceedings of the National Academy of Sciences paper indicates, <a href="http://www.pnas.org/content/early/2014/05/08/1319030111" target="_blank">evidence</a> favors active learning in STEM courses over lecture. So, whether an instructor prefers groups or presentations, if students are engaged, chances are good that they are learning. </div>
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Readers, what classroom organization works for you? </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-72780422848560747572014-05-13T06:58:00.000-07:002014-05-13T06:58:12.353-07:00Building an Effective In-Class Learning Environment, Part 2: Student Presentations<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
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In this series, I explore the questions: What are some advantages and disadvantages of group work and student presentations? How can students be held accountable for learning in groups and from student presenters? What defines a good balance of group time with whole class presentations? In <a href="http://themathswitch.blogspot.com/2014/05/building-effective-in-class-learning.html" target="_blank">Part 1</a>, I focused exclusively on group work.</div>
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In Part 2, I focus mainly on student presentations. Finally, in Part 3, I will discuss considerations involved in balancing time allocated to each of these modes of classroom organization, and balancing the strengths and weaknesses of the two modes against each other.</div>
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<b>Advantages and disadvantages of student presentations to the class:</b></div>
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Student presentations are a way to bring important ideas to the entire class. Presentations focus the entire class on one piece of work. This enables the instructor to monitor the mathematics more easily in comparison to students solving problems in small groups, and to bring up questions to ensure the entire class has the opportunity to grapple with and resolve the key issues in a problem. Student presentations are a good opportunity for the instructor and the class to get an understanding of the presenter’s thinking about a problem. This is especially helpful when a problem has stumped most of the class, so that everyone has a chance to see an idea or tactic that resolves a roadblock. Additionally, individual presentations give the instructor an opportunity to praise a student for sharing his/her thinking about a problem and its solution. Student presentations also enable individual ownership of the mathematics, as the class may later refer back to “Carmen’s solution,” or “Manuel’s way,” etc.</div>
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A major disadvantage of student presentations is that fewer students will participate in a discussion of the solution or proof. This happens not only because the group is larger, but also because many times the audience is afraid to trip up the presenter with a question. Moreover, students are sometimes embarrassed about bringing up their questions in front of the class. Another difficulty is that in a class of more than 20, students sometimes do not work enough outside of class on the problems because of the low probability that they will need to present them. </div>
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<b>Individual accountability during student presentations:</b></div>
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To combat the tendency for fewer students to participate in a discussion of a student presentation, there are a few strategies that can be used.</div>
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<li><b>Call on students in the audience randomly.</b> To ensure equitable participation, call on students randomly. This combats the common problem of having just a handful of students who are willing to comment or ask questions. Students can be asked to paraphrase particular parts of a solution, to identify key pieces of the solution, to identify the type of argument used, or to summarize an entire solution. While it may not increase the number of contributions, calling at random does help to ensure that over the course of a week or so, most students will have a chance to participate in the discussion.</li>
<li><b>Use think-pair-share.</b> One way to generate more discussion is to have students first review the solution/proof on their own, and then pair up to discuss the work of a presenter. Students can be tasked to come up with a question about the presenter’s work, or to provide further explanation for a part of the solution. The instructor then randomly selects some individuals to report on what they discussed with the partner. It is also worth noting that students often have an easier time answering the question, “What did you discuss?” rather than, “What do you think of this solution?” or, “What question do you have?"</li>
<li><b>Let the presenter sit before discussion begins</b>. There can be advantages to letting the presenter moderate the discussion, but if students are shy about putting the presenter on the spot, it may be helpful to let the presenter sit. This does not absolve the presenter from having to answer questions about his or her process in producing a solution, but it often reduces anxiety if the presenter is not standing uncomfortably at the front of the room.</li>
<li><b>Emphasize the importance of discussing ideas, not people</b>. Whether or not the presenter remains in front of the class during discussion of his or her work, it can be helpful to remind the class that suggestions and questions are not personal attacks against the presenter. Instead, emphasize that everyone is learning, and that the presenter would like the feedback now, rather than to find out later that he or she has been making a consistent error. Moreover, if the class finds flaws or makes corrections, the flaws are in the solution, not in the presenter.</li>
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To reduce the tendency of students to spend too little effort outside of class, here are some ideas.</div>
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<li>Although this was also mentioned in the post on group work, <b>check homework at the beginning of class</b> to ensure that individuals already have a record of their own attempts and solutions before discussing their ideas with others. This lets students know that they are being graded for making their own attempts on assigned work outside of class. </li>
<li>Again repeating a suggestion, use <b>colored pens</b> in class. This strategy gives the instructor the power to discern what students are completing on their own time as well as what they are doing in class. As an added benefit over the early homework check, the instructor can encourage students to keep good records in class by commenting on the notes when the assignment is collected, and by giving full credit to assignments that show that all problems were attempted individually AND show corrections and notes that reflect work done in class.</li>
<li><b>Do not accept volunteers for presentations</b>. Typically, at the beginning of a course, it is helpful to let students volunteer. However, shortly thereafter, perhaps by the second week, it is often wise to keep a list of students that have yet to present (and later, the students with the fewest presentations), and to call on those students first. While students will often have significant breaks between presentations, calling on students with the fewest presentations ensures that students know that they are all expected to contribute. </li>
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While all of these strategies reduce the tendency of students to disengage from presentations, I find that in practice, a mix of group work and presentations works best. In the final post in this series, I will examine considerations involved in using a combination of group work and individual presentations.</div>
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Readers, do you have other ideas about how to get the most from students before and during student presentations?</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com2tag:blogger.com,1999:blog-1442561601441696697.post-76954299897737668982014-05-06T06:50:00.000-07:002014-05-06T06:50:35.667-07:00Building an Effective In-Class Learning Environment, Part 1: Group Work<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
In this series, I explore the questions: What are some advantages and disadvantages of group work and student presentations? How can students be held accountable for learning in groups and from student presenters? What defines a good balance of group time with whole class presentations? In Part 1, I will focus exclusively on group work.
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In Part 2, I will focus on student presentations. Finally, in Part 3, I will discuss considerations involved in balancing time allocated to each of these modes of classroom organization, and balancing the strengths and weaknesses of the two modes against each other.
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<b>Advantages and disadvantages of group work:</b></div>
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Groups are an effective way to organize student learning, as described in numerous research articles, such as the <a href="http://rer.sagepub.com/content/69/1/21.abstract" target="_blank">meta-analysis</a> by Springer, Stanne, and Donovan. Groups tend to be most effective when they are smaller, meaning pairs, or groups of three or four people.</div>
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Groups (or pairs) have the advantage of fostering more conversations in a classroom. There are a lot more people speaking at any given time, and there is a lot more back-and-forth exchange of ideas in groups. Generally, more students have the chance to explain their thinking, and they can get more clarification in a small group. Groups also foster more camaraderie and community in a classroom. This is particularly true if group membership is changed regularly, even daily, so that students have the opportunity to work with many different students in the class.
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There are a couple of disadvantages of small group work. One difficulty is that the instructor has many groups to monitor. Groups sometimes do not solve the problems or fully complete the proofs. If there are several different errors or gaps in understanding across groups, it can be difficult to resolve the gaps or errors that arise. Another challenge is that sometimes, if students know that they will be able to work in groups, they may not put in sufficient effort outside of class. Instead, they hope that their partner(s) will have solutions to their problems. </div>
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<b>Individual accountability in groups:</b></div>
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Thus, it is prudent for an instructor to plan for ways to hold individuals accountable for producing work and for understanding the work of the group. What follows are a few ways to promote individual accountability in a class. Note that some of these are more about encouraging students to work outside of class, while others are about ensuring that everyone in the group understands the work produced.
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<li><b>Check homework at the beginning of class</b> to ensure that individuals already have a record of their own attempts and solutions before discussing their ideas with others. This lets students know that they are being graded for making their own attempts on assigned work outside of class. </li>
<li><b>Collect homework at the beginning of class</b>. This works similarly to checking homework, but in order to collect homework, it is important that the work in class does not depend on the homework being collected. I have found that students prefer to have their work handy in class, so that they can compare their own thinking to what is shared in class. This is true even if the homework problems are separate from the class work, as ideas gained in class can sometimes cause students to rethink their homework.</li>
<li>Use <b>colored pens</b> in class. This strategy does not apply only to groups, but again, it gives the instructor the power to discern what students are completing on their own time as well as what they are doing in class. As an added benefit over the early homework check, the instructor can encourage students to keep good records in class by commenting on the notes when the assignment is collected, and by giving full credit to assignments that show that all problems were attempted individually AND show corrections that reflect work done in class. </li>
<li>A <b>jigsaw</b> (described <a href="http://themathswitch.blogspot.com/2013/11/engage.html" target="_blank">previously</a>) is effective in holding individuals accountable. I find that students often work quite hard to ensure that they understand the work of others in their group, because they know they will have to explain that work almost immediately afterwards. </li>
<li><b>Call on individuals at random</b> to report on the conversation or work completed by the group. Since there are typically many groups in a class, calling on groups at random is one way to encourage the groups to stay on task, and to let the instructor get an understanding of the thinking of several students, even when the instructor may not have been able to visit with that group while they were working.</li>
<li>Request <b>group reports</b>. An instructor can hold all group members accountable by asking that every group report its solution to a designated problem to the instructor separately (i.e., not in front of the class). The instructor then queries all members of the group about the solution. For instance, if the class is working on problems 7-12, then all groups may be asked to check in with the instructor when they are satisfied with their solution to problem 8. As the instructor wanders through the room, groups signal when they are ready to share their solution. The instructor then asks a particular group member to begin explaining the solution, stops the explanation to ask others to clarify particular points, or asks others to take over the explanation at that point. In this way, the group must ensure that all members understand the work. If a group member is stumped by an instructor question or gets stuck in an explanation, the instructor tells the group to discuss the work some more and call the instructor back when everyone is ready. </li>
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With appropriate tools in place, groups can be very productive and make for a very lively classroom learning environment. Readers, what other strategies do you use to ensure that groups are effective? </div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-13064608899793028842014-04-22T06:33:00.000-07:002014-04-22T06:33:20.038-07:00Working in the Cloud, Part 3: Evernote<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
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In this third part of the series, I am going to describe how I use Evernote. In earlier posts, I described my use of <a href="http://themathswitch.blogspot.com/2014/04/working-in-cloud-part-1-dropbox.html" target="_blank">Dropbox</a> and <a href="http://themathswitch.blogspot.com/2014/04/working-in-cloud-part-2-google-drive.html" target="_blank">Google Drive</a>.
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Like Google Drive, Evernote has many features beyond data storage. As in my previous two posts, I will describe how I use it, rather than attempt to be comprehensive.
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First and foremost, Evernote is great for typing up notes of all sorts. At this point, I often find that I start drafting up my ideas for lots of things in Evernote. Even if it is eventually going to be an email or a presentation or a blog post, I will usually start with Evernote. I am often taking notes in meetings, or while on the phone. No longer do I have these notes on various papers that I either have to carry around or that I forget at the office when I am somewhere else. I also make notes for my classes, and keep those in a note. Then I pull up the note on any device, and I am ready for class. I use Evernote to save receipts too, particularly if I need to file for reimbursement later.
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Editing notes is the centerpiece of Evernote, so it can be done on any device. Moreover, the interface looks very similar on tablets, laptops, and the desktop, so I don’t have to spend a lot of time finding the features I use on one device when I move to a different one.
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Notes are accessible in a few different ways. I can add tags to my notes, and later pull up notes with a particular tag or set of tags. Notes can also be sorted into different notebooks, which are the equivalent of file folders in other storage systems. Notes that have reminders will be listed under reminders. For a note that I return to frequently, I add it to a list of shortcuts. Finally, notes are sorted by date of creation, with the most recent notes at the top within easy reach. This makes organization, and therefore finding the note I want to work on, pretty easy.
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One of my favorite ways to use Evernote is in conjunction with its native reminder feature. For instance, if I need to photocopy something at the office, I might type a quick note to myself in Evernote and set up a reminder. The reminder can have a due date or not. If it does have a due date, Evernote sends an email to you on that day as well.
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Have you ever had the experience of having an email you can’t get to right now but want to answer later? This is where Evernote works really well. Evernote accounts come with an email address. Suppose I have an email from Maria that I want to get to later. I will forward Maria's email to my Evernote address. When it arrives, I add a reminder to the email-now-note, and then I do not forget to respond (reminders can also be added by altering the subject line of the email when I forward it). There are also times when I receive an email request that I can respond to, but also requires additional action later. When I respond, I blind-copy my Evernote address, so that I see the email and it reminds me to follow up on it.
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In fact, Evernote interacts well with email in other ways. You can type a quick note as an email, and send it to Evernote. You can even set it up so that the email appends to an existing note, gets filed in a particular notebook, is tagged the way you want it, and it can have a reminder added as well.
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Another way that I have come to use Evernote is to save items from the web, in a few different ways. Evernote has a web clipper for your task bar that lets you quickly take a screenshot (entire screen or a selected portion) and save it to Evernote. This works well if there is a graph or some data in a table that I want to save. If I am reading a blog post, often through Feedly or Instapaper, and I decide I want to keep the article, then I save it to Evernote. Then I tag it, and in this way I build a collection of ideas to share with colleagues, with students, with teachers, etc.
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I also share my notes in one of two ways. Either I email the entire note, or I get a URL for the note, and share that instead. An advantage of using the URL is that I can post it somewhere, like a Lino or Padlet board, or some other public web space, and anyone can follow the link to see the note. Another advantage is that by following the URL, anyone accessing the note will see the most updated version.
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Evernote’s storage limit works differently than Dropbox or Google Drive. Rather than a fixed limit, Evernote allows its users to <i>add</i> up to 60 MB (for the free account) per month. So I don’t worry that I have to go back through old notes to delete them; rather, I just have to be careful of how much I add each month (and I have never hit my limit).
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Now we come to limitations. Evernote is great for notes. I also add PDF files, scans of documents, and occasional photos to Evernote. But I do not save other sorts of files, like spreadsheets or LaTeX files. I could save a LaTeX file into Evernote, but I am not sure that I could go through my usual cycle of editing, compiling, editing more, etc. Files other than notes are saved by "clipping" them to a note. Evernote doesn't seem made for the purpose of keeping files that are updated a lot. In this sense, Evernote probably could not be my only cloud service.
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Syncing with Evernote is reliable, but not constant. Whereas every change is sent to Google Drive every few seconds as I work, and Dropbox is always working in the background to sync my files once saved, Evernote syncs either when prompted, or at other intervals. This has only come up when I make a last-minute addition to a note for my class in Evernote on my desktop, and then proceed to class with Evernote on a different device. If I don’t hit the sync button on Evernote, it may be a while before it syncs, and therefore I am without the latest version of my notes for class.
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Basic (read: free) Evernote accounts require internet access in order to access my notes. This makes it the same as Google Drive. However, because Evernote is not syncing my notes every few seconds, if I am on a slow connection, I am not as likely to lose access to my note. However, I will lose the ability to open other notes if I am without internet. Evernote Premium users can access notes offline by designating particular notebooks that will be stored locally on the device.
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Evernote Summary:
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Advantages:
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+Notes can be created and edited on any device.
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+Notes can be organized in multiple ways, including notebooks, tags, reminders, and shortcuts.
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+Notes can be added via email.
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+Notes can have reminders, with or without dates.
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+Notes can be created by saving items from the web.
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+Evernote has an upload limit per month, rather than a fixed storage limit.
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Disadvantages:
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-Evernote is not designed for editing spreadsheets or PDFs or other files besides notes.
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-Sync immediately before moving to another device to guarantee notes are up-to-date.
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-For Basic users, accessing notes requires an internet connection.
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There are certainly a lot more ways to use Dropbox, Google Drive, and Evernote than I have covered here. Still, I hope this has given readers at least an idea of some of the different ways of taking advantage of these three services. </div>
Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-15569171573478177072014-04-15T09:44:00.000-07:002014-04-15T09:46:43.287-07:00Working in the Cloud, Part 2: Google Drive<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
In my <a href="http://themathswitch.blogspot.com/2014/04/working-in-cloud-part-1-dropbox.html" target="_blank">last post</a>, I covered my usage of Dropbox. In this post, I am going to pick up with my usage of Google Drive.</div>
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When I only had home and office desktops and a laptop, I was satisfied with Dropbox alone. Two years ago, I got an iPad, and began to explore more apps, including Evernote and Google Drive. Given that I was already a Dropbox user, I have added the other services to fulfill particular needs. If I had come to the others first, my usage of them would probably be different. In this post, I will discuss Google Drive.</div>
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Whereas Dropbox is really about file storage, Google Drive is a more expansive system. You can use Google Drive to store files of all types. However, files in Google Drive work especially well with Google’s Documents, Sheets, Forms, Presentations, and Drawings. For me, Google Drive serves to enable collaboration in ways that are difficult with Dropbox. In particular, if I have a document that I am going to co-author, and I want my co-author to be able to view the document with me simultaneously, then I will use Google docs. I also like the feature that allows me to select what people can do with an item I share, where the options are: can view/can comment/can edit. So when I am running a workshop, for instance, my co-facilitators may have editing capability, while participants get viewing capability. And, unlike Dropbox, only files that originate with me count against my file storage limit.</div>
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My favorite feature of Google Drive is Google Forms. Google Forms are great for surveys or quizzes. There are several different question formats that you can set up for a question, including multiple choice and short answer. The responses to the form are collected in a Google Sheet. And, with the use of Flubaroo, an add-on to Sheets, I can auto-score a quiz as well, and have the scores emailed to the students.</div>
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There are a couple of nice aspects of storage in Google Drive. One is that Google’s native formats (Documents, Sheets, Forms, Presentations, and Drawings) do not count against your storage limit. Another is that items scanned in to Drive get Optical Character Recognition (OCR) applied to them, so that if, for instance, I scan a hard copy of a typed document, I am saved from re-typing it, because OCR converts the scanned content into text.</div>
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Google Drive has its own set of limitations. One of the limitations across Google’s native Docs and Forms is that it is difficult to typeset mathematics. (I have a partial work-around, but I’ll save that for another day.) Another issue is that Documents and Sheets are editable on the iPad, but at last check, Drawings and Forms are not. Also, where Dropbox files are stored locally on a laptop or desktop, so that you can work offline, and particular files can be selected for local storage on your tablet, Google Drive files are not generally available offline. I have had the experience of trying to access a file when I have a slow internet connection, and I am stuck unable to access it. (If you use Chrome or Chrome OS, you can set up offline access: <a href="https://support.google.com/drive/answer/2375012?hl=en">https://support.google.com/drive/answer/2375012?hl=en</a>.)</div>
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Google Drive Summary:</div>
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Advantages:</div>
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+Native editor for proprietary file types.</div>
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+Native Document and Sheet editors work on the iPad.</div>
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+Forms are great for surveys and quizzes. </div>
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+Form results go in a spreadsheet that can be auto-scored with Flubaroo.</div>
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+Files in native Google formats do not count against the storage limit.</div>
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+Only my own files count against my storage limit, and not those files shared with me but owned by others.</div>
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+OCR lets me convert hard copies into electronic text.</div>
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-Mathematics is difficult to typeset in Google’s native formats.</div>
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-Google Forms, Presentations, and Drawings cannot be edited on an iPad.</div>
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-Google files are not available offline except via Chrome or Chrome OS.</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-81811208252001890992014-04-08T06:49:00.000-07:002014-04-08T06:49:43.229-07:00Working in the Cloud, Part 1: Dropbox<!--?xml version="1.0" encoding="UTF-8" standalone="no"?--><span style="font-family: Arial, Helvetica, sans-serif;">
I use multiple cloud services, primarily Dropbox, Google Drive, and Evernote. This has been my setup for the past year or so, whereas before that, I used Dropbox exclusively. In this series of posts, I’d like to explore what I like and dislike about each service, and how I have divided my work amongst them. Please note that I am not attempting to be comprehensive, but rather to capture what I have found helps or hinders my own workflow.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I discovered Dropbox a few years ago. Before that, I transported important files between work and home using a flash drive. However, I occasionally forgot the drive at home and had to improvise, either by having someone at home email me a file, or by re-creating work, or simply not working on a project that day at the office. Dropbox solves this problem by letting me access files from anywhere. Dropbox is, in part, a special file folder on my computer. If I designate a computer to have Dropbox installed, then all of the files that are in my Dropbox folder will be uploaded and synced to Dropbox, so that the files automatically update between, for instance, my desktop computer at home and my desktop at work. I can also access the files on my iPad, iPod, Samsung tablet, etc. On tablets and phones, Dropbox does not store most of the files on the hard drive by default; instead, it downloads whichever one I want to open when I want it. However, I am also able to designate favorites, which means Dropbox stores a copy locally on the tablet/phone drive. In this way, Dropbox has worked well.
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<span style="font-family: Arial, Helvetica, sans-serif;">In addition, there are a few ways to collaborate and share files with others. I can share a link to a file or folder, or I can create a shared folder. A link to a file allows someone else with the file link to view and download the file. A shared folder lets me share an entire folder with another person, with shared editing rights. I have used these options often when I am running a workshop, and I want participants to have access to multiple files that will be used during the workshop.
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<span style="font-family: Arial, Helvetica, sans-serif;">One difficulty with shared folders is that edits made by one person affect everyone. This is especially noticeable when one person drags a file from the Dropbox folder to another location on their hard drive. The person is effectively deleting the file from the Dropbox shared folder, and as a result, deleting the copy of it from everyone else’s shared folder as well. I have learned that shared folders are difficult to manage for this reason.
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<span style="font-family: Arial, Helvetica, sans-serif;">Another limitation of Dropbox is that it is not designed for simultaneous editing on multiple devices. If I leave a file open on one computer, and then I edit it on another device, I will get two copies of the file as a result. Edits are not integrated into an open copy on another machine.
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<span style="font-family: Arial, Helvetica, sans-serif;">Also note that as with all cloud storage systems, there is a storage limit. If you have a shared folder, the entire folder counts against your limit. Thus, last year, when someone wanted to share a very large folder with me, by choosing to share in that folder, I overran my free space. So I either had to un-join the shared folder, or begin a paid Dropbox plan (I chose the latter).
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<span style="font-family: Arial, Helvetica, sans-serif;">Another limitation for Dropbox is that there is no native editor. This is not a major limitation, but it can be a nuisance to have to open files with particular apps, especially on a tablet. This can lead to formatting errors when files edited in one app on the desktop are edited with a different app on the tablet.
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<span style="font-family: Arial, Helvetica, sans-serif;">Dropbox Use Summary:
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<span style="font-family: Arial, Helvetica, sans-serif;">+Files are stored locally on the hard drives of my desktops and laptop.
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<span style="font-family: Arial, Helvetica, sans-serif;">+Files can be designated for local storage on tablets and phones.
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<span style="font-family: Arial, Helvetica, sans-serif;">+Files can be shared for download by others.
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<span style="font-family: Arial, Helvetica, sans-serif;">+Shared work folders enable collaboration and shared editing.
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<span style="font-family: Arial, Helvetica, sans-serif;">Disadvantages:
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<span style="font-family: Arial, Helvetica, sans-serif;">-Files removed from shared folders are removed from all devices.
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<span style="font-family: Arial, Helvetica, sans-serif;">-A single file cannot be edited by multiple users simultaneously. The file will be split into copies.
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<span style="font-family: Arial, Helvetica, sans-serif;">-Shared folders count against the storage limit of all participants.
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<span style="font-family: Arial, Helvetica, sans-serif;">-There is no native ability to edit files of any sort.
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<span style="font-family: Arial, Helvetica, sans-serif;">Next time, I will describe how I use Google Drive.</span></div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-15809103172785700832014-03-25T10:14:00.000-07:002014-03-25T10:14:06.586-07:006 Ways for Students to Grapple with Definitions<!--?xml version="1.0" encoding="UTF-8" standalone="no"?--><span style="font-family: Arial, Helvetica, sans-serif;">
This is a relatively short post. To prepare for workshops I am running for college math faculty this summer, I am thinking about the skills that an instructor needs to be able to be effective in using IBL. One of the big skills is building problem sets for students, and within that, one of the component skills is being able to help students develop an understanding of definitions. </span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">Specifically, I am thinking about how I try to get students to begin processing the ideas in a definition, before putting their understanding to the test with problems or theorems that require application of the definition. What follows are 6 ways that I have thought of, so far, that get students to begin to attach meaning to a definition. </span><div>
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<span style="font-family: Arial, Helvetica, sans-serif;">For purposes of illustration, I will refer to the definition of prime. A natural number p is <b>prime</b> if and only if p is not the product of natural numbers less than p.
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<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Have students sort a list of candidates into examples and non-examples.</b> Prime: Which of the numbers -3, -1, 1, 2, 3, 4, 5 are prime?</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Fill in the blank with all, some, or none. </b>Prime: All/some/none of the even natural numbers are prime. All/some/none of the odd natural numbers are prime. </span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Prove a very basic and concrete example.</b> Prime: Prove that 3 is prime, using the definition. Prove that 4 is not prime, using the definition.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Ask students to construct their own examples.</b> Prime: List 5 prime numbers. (Notice that I chose 5 here to force students to confront that 9 is odd, but not prime.)</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Relate the current definition to concepts encountered earlier.</b> Prime: We have been discussing divisibility in this course. Rewrite the definition of prime using forms of the word “divisibility” where appropriate.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;"><b>Pose a true or false question about the definition.</b> Prime: True or false: 2 is the only even number that is prime.</span></li>
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<span style="font-family: Arial, Helvetica, sans-serif;">The main idea in all of these is to get students to unravel the definition by learning to distinguish the important features of the definition, especially which objects are included and which are excluded. I do not use all of these methods with every definition, but rather use a couple of them for each new term. Depending on students’ familiarity with the specific definition, asking students to construct their own examples is typically more difficult than the other tasks. Also, non-examples almost certainly must be supplied by the instructor, since a student, by definition just learning about the idea, is unlikely to have an idea of what kinds of objects are useful non-examples. For instance, a ballpoint pen is not a prime number, but this is not a useful non-example! However, note that students <i>can</i> be asked to determine whether a mathematical object is or is not an example of the definition being introduced.
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<span style="font-family: Arial, Helvetica, sans-serif;">I would love to add to this list. Readers, please share your suggestions!
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com2tag:blogger.com,1999:blog-1442561601441696697.post-30169925541325676672014-02-25T14:38:00.000-08:002014-02-25T14:38:01.084-08:00The instructor's role in an IBL class, Part 2<div style="-webkit-text-stroke-width: 0px; color: black; font-family: Arial; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px;">
In my <a href="http://themathswitch.blogspot.com/2014/02/the-instructors-role-in-ibl-class-part-1.html" target="_blank">last post</a>, I described three of five aspects of an IBL instructor’s role: managing expectations, managing emotions, and keeping the students engaged. In this post, I take up the remaining aspects.</div>
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<b>Finding out what students know</b> is an ongoing task. For those who follow such things, this is also called formative assessment, and it is a critical part of a successful teaching-and-learning experience. There are a couple of purposes for formative assessment. One is to formulate responses as the instructor that will help students move forward in understanding the mathematics. Another is to identify opportunities where specific students may benefit from working on particular problems, or to find opportunities for students to share what they know at a time when it will benefit the class. One of the great benefits of teaching via IBL is that there are so many opportunities to hear from students and to develop a picture of where they are in their mathematical development. By listening to discussions between and among students in pairs or groups, and during presentations and the ensuing discussions, the instructor should have a good idea of when students might have something especially productive to contribute, or when a discussion from one group should be shared with the whole class, for instance. Notice that while formal quizzes or exams remain a source of information, as an IBL instructor the opportunities to find out what students are thinking go far beyond this, and are embedded in the everyday tasks of the class. Also notice that grades are not really a purpose of formative assessment. The focus is on student learning, and how to enhance it.</div>
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<b>Fitting the problems to the students</b> is a task that begins before the semester, but continues to occur through the semester. Before the semester, the major task of an IBL instructor is to determine the main course content goals, which could be particular theorems, skill with specific kinds of problems, or facility with certain techniques. Sources for beginning this work on your first attempt with a class might be the department course syllabus, and/or standard textbooks. From these, the instructor’s job is to put a priority on the central ideas. Then, the instructor works on developing a sequence of problems, lemmas, etc., that will carry the students from their anticipated starting point through to the goal results. As the semester gets underway, the IBL instructor works (1) to find problems to engage particular students (often the highest students or the ones struggling the most), or (2) to use to the students' advantage what they know and are thinking about, and to respond with a set of problems that provide an alternate path to the results, and (3) to modify the difficulty of the problems as the students may be more or less advanced than anticipated and more or fewer lemmas are needed between the main results to keep the majority of the class moving in a positive direction. </div>
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I hope this captures at least some of the key ingredients in the recipe for a successful IBL course. Let me know your thoughts.</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com0tag:blogger.com,1999:blog-1442561601441696697.post-31034846750429865412014-02-17T12:07:00.000-08:002014-02-17T12:07:05.821-08:00The instructor's role in an IBL class, Part 1<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">In discussing inquiry-based learning (IBL) with college faculty and K-12 teachers, I find that one of the difficult things to do is to communicate what the instructor’s role </span><b style="font-family: Arial;">is</b><span style="font-family: Arial;">, as opposed to what it is not. Many people are familiar with such mantras as, Teaching is not telling, or, Don’t lecture. These are helpful, but then instructors are left wondering what to do. In this post, I want to briefly describe a few important duties of an instructor in an IBL classroom. The roles I am going to describe are not mutually exclusive categories, but interwoven threads. Nonetheless, I call these out because I think they capture some critical aspects of the flavor of teaching an IBL course. These duties are: managing expectations, managing emotions, keeping the students engaged, finding out what students know, and fitting the problems to the students. In this post, I will deal with the first three aspects, and deal with the last two in my next post.</span><br />
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<b>Managing expectations</b> is a primary duty in an IBL classroom. Students come to class, and especially, come to math class, with expectations, including unconscious ones, about what is going to happen. These expectations are often something like, "The teacher will show me a formula and examples, and I just have to memorize and repeat what the teacher does on similar examples." In contrast, in an IBL classroom, students are expected to bring their ideas to problems for which the path to solution may not be clear. Students are not used to being asked to think things through for themselves in math class, and this leads to frustration. The teacher’s first duty is to make it clear to students that they will need to bring their own ideas, and that they will often not know what to do, or they will do things that turn out not to work, but as a class, they will make progress in understanding the mathematics. In class, the IBL instructor can say things like, “This is going to be different, but you will learn a lot,” or, “You’re going to experience mathematics the way that mathematicians do,” or, “You will get stuck a lot in this class. That’s ok. You can even write ‘STUCK!’ on your work when that happens. The important thing is to learn from what you try, both what works and what doesn’t."</div>
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In tandem with managing expectations is <b>managing emotions</b>. As mathematicians, we experience frustration as we search for a solution, and we take wrong turns, or the path to the solution is longer than we hoped. Students feel this frustration. If you are managing expectations properly, then students should know that frustration is normal and expected. However, there is more to the instructor’s role than that. If the entire class is boiling over with frustration, the instructor has a duty to respond. If the students are left to flounder, a mutiny can begin to brew. The instructor may say things like, “It seems like this problem/theorem is really stumping us. Let’s brainstorm how we can find new ways to attack it,” or, “I am glad to see everyone is showing persistence on this problem. Sometimes the best way to get past a roadblock is to go around it. So why don’t we look at this {example, related theorem, special case} for now and then come back to the main problem,” or, “This problem is really giving us a rough ride. Let me tell you a quick story about this time when I was frustrated and how I got through it…”</div>
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<b>Keeping the students engaged</b> is a multifaceted task. I have written on this blog before about a <a href="http://themathswitch.blogspot.com/2013/11/engage.html" target="_blank">specific</a> engagement strategy, and about <a href="http://themathswitch.blogspot.com/2013/08/9-ways-to-engage-reluctant-students-aka.html" target="_blank">how to reach out to students who are reluctant</a> to participate. I will add to what I’ve written before by mentioning the importance of managing classroom participation. When there is a student presenting work to the class, the instructor’s role is to ensure that the rest of the class is engaging with the presenter and/or the presenter’s work. This can be done in several ways. One of them is to give the students question/comment stems and to call on students to use the stem to offer a question or comment. Some of my stems include, “I like how you…” “What led you to think of…” “I’m not sure I follow how you got from … to …” Another way is to have the presenter pause and to do a check for understanding or a think-pair-share. When the presentation is done, the instructor then pushes the students to ensure that they agree with all aspects of the mathematics, and also see to it that they understand insofar as possible how the student came up with that solution. When students are working in pairs or small groups, the instructor should make sure that all group members are contributing to the conversation. When necessary, this can be actively managed when the instructor inserts him or herself into a group conversation to make statements such as, “I see that Ann and Bob are sharing their ideas. Carol, is what they’re saying making sense to you?” or, “I appreciate that you’re all giving Darryl your attention. Could someone else try to restate what Darryl has shared so far?” or, “It seems like this group has spent some time working independently, each person with his or her own ideas. I think now would be a good time to share some of your progress and see what you can learn from each other’s approaches to this problem."</div>
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In the next post, I will take up the other two aspects of the instructor's role.</div>
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Let me know your thoughts or other aspects of the instructor's role I have left out.</div>
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Anonymoushttp://www.blogger.com/profile/07891267615270088161noreply@blogger.com1tag:blogger.com,1999:blog-1442561601441696697.post-47488354455732920842013-12-06T14:27:00.000-08:002013-12-06T14:27:11.337-08:00Formative Assessment<!--?xml version="1.0" encoding="UTF-8" standalone="no"?-->
<span style="font-family: Arial;">An idea that has been coming up a lot in several different contexts for me is formative assessment. Let me start by stating what formative assessment is, and what it is not. Formative assessment is information gathered from students to assess their current understanding, with the purpose of using that information to make instructional decisions. Formative assessment need not be a formal exam, and cannot be an exam if it will not impact instruction after the exam. Thus, when I am speaking with folks in school districts, sometimes I say “formative assessment” and this is interpreted to mean something like quarterly (or whatever frequency) benchmark tests. In fact, my experience is that teachers rarely are given the time or resources to use benchmark tests as formative assessment. Instead of informing further instruction, they disrupt instruction, as teachers interrupt their regular lessons to review for the benchmark exam, and after the exam, hurry to move on to whatever is next on the overstuffed curriculum pacing guide. As described here, benchmark tests are NOT formative assessment. Instead, when I think of formative assessment, I think of day-to-day tasks that allow the teacher to gather information about what students know, and give the teacher the chance to address gaps and other issues students are having in understanding the ideas of the course.</span><br />
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This semester, one of the best things I learned was how to build questions that would serve as good formative assessment. The questions I have been using were described in my earlier post discussing how I deal with <a href="http://themathswitch.blogspot.com/2013/10/dealing-with-misconceptions-part-2.html" target="_blank">misconceptions</a>. Since this post is about formative assessment, I want to describe how I react to students’ responses to the questions. The questions I have been using are frequently true-false questions, or sometimes multiple choice. Students are first given time to respond to the questions alone (most commonly I have been using Google Forms to collect their initial response), and then to discuss their responses with their peers. Since the discussion may alter their opinions, I then ask for a show of hands for each answer choice. I have seen a few things happen.
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<li>Sometimes there seems to be broad consensus on the correct answer. In that case, I will ask one or two students to summarize the reasons for the correct choice, record the answer for the class, and move on.</li>
<li>Sometimes, the hand votes are close to equally split between two choices. In this case, I try to get at least one person on each side to articulate the reasons for their answer choice. Then I either ask follow-up questions or I ask other students to add to the arguments for each side. Sometimes this is enough for students to see which is the correct choice. I hear students saying things like, “Oh, I didn’t think of that example,” or, “I changed my mind.” If I feel that there is consensus, then I will record the correct answer at the board and summarize the discussion. If the discussion is not progressing, then I usually prompt students to come up with one or more examples or to draw a graph or diagram related to the statement. Since the topics in these questions are not new, students generally have enough knowledge to resolve the questions. Lastly, I may refer them back to previous work that we did, or pull up the work of a student from the previous class meeting. One of these moves is generally enough to push the discussion toward the correct answer.</li>
<li>The third thing that sometimes happens is that I see very few hand votes. I generally take this as a sign of confusion. In that case, I will do one of two things. Sometimes, I tell the students that I see very few votes, and that they need to go back to the discussion with their partner for a couple more minutes to settle things before we can have a class discussion. Then we vote again. More typically, I call on students who raised their hand to explain their thinking, and then call on those who did not vote to see if they are following the argument. After enough students have participated, and I am satisfied that the main ideas have been discussed and reiterated sufficiently, I will ask students for any final questions, and then summarize the discussion and record our answer. </li>
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Formative assessment can be a powerful tool. How do you assess students and use that to inform your instructional moves?</div>
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