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?

Building an Effective In-Class Learning Environment, Part 1: Group Work

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.

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.

Advantages and disadvantages of group work:

Groups are an effective way to organize student learning, as described in numerous research articles, such as the meta-analysis by Springer, Stanne, and Donovan. Groups tend to be most effective when they are smaller, meaning pairs, or groups of three or four people.

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.

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. 

Individual accountability in groups:

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.

1. 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. Collect homework at the beginning of class. 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.
3. Use colored pens 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. 
4. A jigsaw (described previously) 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. 
5. Call on individuals at random 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.
6. Request group reports. 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. 

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?