Teach Like a Freak, Part 2

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.

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.

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. 

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.

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.

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.

So, to my fellow educators, get your freak on! Try new ideas, and tackle those small problems.

Teach Like a Freak, Part One

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.

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 (http://www.nctm.org/publications/article.aspx?id=42527) and active learning more generally (http://www.pnas.org/content/111/23/8410.abstract) are more effective than lecture across multiple outcomes. At the same time, I have been reading research about Interactive Engagement (http://www.ams.org/notices/201308/rnoti-p1018.pdf), 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. 

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.

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.

In part 2, I will take up two other ideas from Think Like a Freak, targeting small problems and thinking like a child. 

6 Responses to Students' Questions About IBL

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, 

"Why do we have to teach ourselves?" "I don't learn this way," or, "Why not just show me what to do?"

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.

1. 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.
2. 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 Marzano, Pickering, and Pollock, 2001, for example. The recent article by Freeman and colleagues reviews research on active learning at the college level.)
3. 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 (http://ptjournal.apta.org/content/81/3/896.full).
4. 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.
5. 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.
6. 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.

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.  

Who counts?

In this post, I discuss federally defined graduation rates. 

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. 

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.

Much of the discussion of graduation rates is based on the Federal Graduation Rate (FGR), which measures the percentage of first-time, full-time freshmen who graduate within six years of entering their original four-year institution. 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.

For purposes of illustration, suppose we are looking at the graduation rate for “Big State University (BSU).” In the FGR:

• 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!
• 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.
• 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.

Taking these factors into account, the American Council on Education estimates that about 61% of students at 4-year schools are excluded from the calculation. Sixty-one percent! How do institutions have a conversation about their successes and shortcomings when they ignore more than half of the students in their calculation?

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.

Using public Integrated Postsecondary Education Data System (IPEDS) data from my institution, the data and my calculations indicate that 306 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 2,481 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?

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.

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.

NOTE:

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.

Here is one example of an article discussing some of these issues:

http://www.acenet.edu/the-presidency/columns-and-features/Pages/Why-Graduation-Rates-Matter—and-Why-They-Don’t.aspx

The Excitement of September

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.

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. 

I hope your year is off to a good start. 

What is acceptable evidence?

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.

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? 

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 explain operations on fractions using models such as the area model and the number line, apply the models in realistic contexts, solve problems involving fractions, and interpret their answers.

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. 

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.

Have you worked with your goal statements? Do you find that you assess progress toward your goals regularly? 

Examining Reasons to Use Technology in the Classroom: Mathematical Modeling of Flight Times

In this post, I explore how technology has made mathematical modeling more accessible.

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.

As I have written previously, 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. 

In that problem, we started by making observations about a map produced by http://www.flighttimesmap.com 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. 

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.

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.

Stepping back from the problem, here is a look at how technology enhanced this exploration.

• 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).
• 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.
• 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.
• 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.

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.

Keep Tinkering

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.

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.

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.

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. 

Most popular posts from one year of blogging

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:

1. 9 Ways to Engage Reluctant Students, aka Tackling the Startup Problem 
2. Harnessing Your Personality 
3. Dealing with misconceptions, Part 1: Seven ways to handle misconceptions in the moment 
4. Engage! 
5. A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins 

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. 

An Introduction to 6 Apps for Quizzes and Polls

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.

Readers may also wish to consult the comparison chart at http://www.polleverywhere.com/vs (the chart dates from November, 2012), and to look at some of the information provided by Richard Byrne at http://www.freetech4teachers.com/2013/03/four-good-alternatives-to-clicker.html and elsewhere on his site.

1. Edmodo: https://www.edmodo.com 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.
2. Socrative: http://socrative.com 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 without 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). 
3. Google Forms 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 earlier). 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.
4. Quiz Bean 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.
5. Quizlet 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. 
6. gFlash+ 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.

Besides the ones listed above, there are many, many more. An incomplete list of them includes:

1. Exit Ticket: http://exitticket.org
2. Kahoot: https://getkahoot.com
3. Mentimeter: https://www.mentimeter.com
4. ParticiPoll: http://www.participoll.com
5. Poll Everywhere: http://www.polleverywhere.com. 
6. TAPit: http://theanswerpad.com
7. Flisti: http://flisti.com
8. Infuse Learning: http://www.infuselearning.com
9. Quiz Socket: http://www.quizsocket.com
10. Geddit: http://letsgeddit.com
11. Top Hat: https://tophat.com

I hope this spurs some ideas. There are so many ways to collect feedback from students!

Doing math on iOS

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.

Here is the list of iOS apps I have tried for doing math of various sorts on my iPad:

• 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).
• 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.
• 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.
• 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 on 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.

The following are apps that I have used, but not extensively:

• 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.
• 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.
• 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.

Readers, what have I missed?

Taking notes on iPad: Evernote, NotesPlus, and Notability

In this post, 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.

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 Evernote. 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.

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 Notes Plus or Notability for class notes. I can recommend both apps. For NotesPlus and Notability:

• Both can be backed up to Dropbox. 
• Both offer an eraser as well as an undo button. 
• Both offer a close-up box for writing. 
• Both apps have always retained everything I’ve created. I have never experienced disappearing notebooks or pages.
• Both apps offer a variety of pen thicknesses and colors as well as a highlighter.
• Both apps offer the ability to add audio recordings to notes.
• 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. 
• 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. 
• 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. 

Together, this set of tools has really helped me get the most productivity from my iPad.

9 Books to Read and Reread

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. 

For any teacher, I recommend:

• What Works in Schools? Robert J. Marzano, Debra J. Pickering, and Jane E. Pollock. (Note that there is now a second edition 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.
• Why Don’t Students Like School? 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.
• What’s the Point of School? 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.
• Switch. 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.
• Understanding By Design. 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.
• Mindset. 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.

For math teachers at any level, I recommend:

What’s Math Got To Do With It? 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.

For college teachers, I recommend:

What the Best College Teachers Do. 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.

Finally, for college math teachers, I recommend:

The Moore Method: A Pathway To Learner-Centered Instruction. 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.)

What are some of your favorite or most inspiring reads from the educational realm?

Typesetting Mathematics in Google Drive

In a previous post, 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. 

Options for creating images with mathematical symbols are:

1. 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. 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 SciWeavers. 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.
2. 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.
3. 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.
4. g(Math) 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). 

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.

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.

After I composed this post, I wanted to make sure everything was up-to-date, and I found this thread https://productforums.google.com/forum/#!topic/docs/mMQl4IkKG2c that includes the same suggestions for Forms. 

Joys of Teaching

In this post, I explore the question: What are the joys of teaching?

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.

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.

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.

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.

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! 

Feel free to share your favorite joys of teaching in the comments. 

Building an Effective In-class Learning Environment, Part 3: Balancing Group Work And Student Presentations

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 focused exclusively on group work.

In Part 2, 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. 

Striking a balance between group work and student presentations:

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.

Given the challenges and opportunities associated with group work and student presentations, what mix of these two forms of classroom organization is best?

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.

Group-centered model:

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.

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.

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.

Presentation-centered model:

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 in physics 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 definitions or examples that may help them.

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.

Final comments:

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, evidence 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. 

Readers, what classroom organization works for you? 

Building an Effective In-Class Learning Environment, Part 2: Student Presentations

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 focused exclusively on group work.

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.

 

Advantages and disadvantages of student presentations to the class:

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.

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. 

Individual accountability during student presentations:

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.

1. Call on students in the audience randomly. 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.
2. Use think-pair-share. 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?"
3. Let the presenter sit before discussion begins. 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.
4. Emphasize the importance of discussing ideas, not people. 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.

To reduce the tendency of students to spend too little effort outside of class, here are some ideas.

1. Although this was also mentioned in the post on group work, check homework at the beginning of class 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. 
2. Again repeating a suggestion, use colored pens 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.
3. Do not accept volunteers for presentations. 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. 

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.

Readers, do you have other ideas about how to get the most from students before and during student presentations?