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.  

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. 

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. 

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?

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? 

6 Ways for Students to Grapple with Definitions

This is a relatively short post. To prepare for workshops I am running for college math faculty this summer, I am thinking about the skills that an instructor needs to be able to be effective in using IBL. One of the big skills is building problem sets for students, and within that, one of the component skills is being able to help students develop an understanding of definitions. 

Specifically, I am thinking about how I try to get students to begin processing the ideas in a definition, before putting their understanding to the test with problems or theorems that require application of the definition. What follows are 6 ways that I have thought of, so far, that get students to begin to attach meaning to a definition. 

For purposes of illustration, I will refer to the definition of prime. A natural number p is prime if and only if p is not the product of natural numbers less than p.

1. Have students sort a list of candidates into examples and non-examples. Prime: Which of the numbers -3, -1, 1, 2, 3, 4, 5 are prime?
2. Fill in the blank with all, some, or none. Prime: All/some/none of the even natural numbers are prime. All/some/none of the odd natural numbers are prime. 
3. Prove a very basic and concrete example. Prime: Prove that 3 is prime, using the definition. Prove that 4 is not prime, using the definition.
4. Ask students to construct their own examples. Prime: List 5 prime numbers. (Notice that I chose 5 here to force students to confront that 9 is odd, but not prime.)
5. Relate the current definition to concepts encountered earlier. Prime: We have been discussing divisibility in this course. Rewrite the definition of prime using forms of the word “divisibility” where appropriate.
6. Pose a true or false question about the definition. Prime: True or false: 2 is the only even number that is prime.

The main idea in all of these is to get students to unravel the definition by learning to distinguish the important features of the definition, especially which objects are included and which are excluded. I do not use all of these methods with every definition, but rather use a couple of them for each new term. Depending on students’ familiarity with the specific definition, asking students to construct their own examples is typically more difficult than the other tasks. Also, non-examples almost certainly must be supplied by the instructor, since a student, by definition just learning about the idea, is unlikely to have an idea of what kinds of objects are useful non-examples. For instance, a ballpoint pen is not a prime number, but this is not a useful non-example! However, note that students can be asked to determine whether a mathematical object is or is not an example of the definition being introduced.

I would love to add to this list. Readers, please share your suggestions!

The instructor's role in an IBL class, Part 2

In my last post, I described three of five aspects of an IBL instructor’s role: managing expectations, managing emotions, and keeping the students engaged. In this post, I take up the remaining aspects.

Finding out what students know is an ongoing task. For those who follow such things, this is also called formative assessment, and it is a critical part of a successful teaching-and-learning experience. There are a couple of purposes for formative assessment. One is to formulate responses as the instructor that will help students move forward in understanding the mathematics. Another is to identify opportunities where specific students may benefit from working on particular problems, or to find opportunities for students to share what they know at a time when it will benefit the class. One of the great benefits of teaching via IBL is that there are so many opportunities to hear from students and to develop a picture of where they are in their mathematical development. By listening to discussions between and among students in pairs or groups, and during presentations and the ensuing discussions, the instructor should have a good idea of when students might have something especially productive to contribute, or when a discussion from one group should be shared with the whole class, for instance. Notice that while formal quizzes or exams remain a source of information, as an IBL instructor the opportunities to find out what students are thinking go far beyond this, and are embedded in the everyday tasks of the class. Also notice that grades are not really a purpose of formative assessment. The focus is on student learning, and how to enhance it.

Fitting the problems to the students is a task that begins before the semester, but continues to occur through the semester. Before the semester, the major task of an IBL instructor is to determine the main course content goals, which could be particular theorems, skill with specific kinds of problems, or facility with certain techniques. Sources for beginning this work on your first attempt with a class might be the department course syllabus, and/or standard textbooks. From these, the instructor’s job is to put a priority on the central ideas. Then, the instructor works on developing a sequence of problems, lemmas, etc., that will carry the students from their anticipated starting point through to the goal results. As the semester gets underway, the IBL instructor works (1) to find problems to engage particular students (often the highest students or the ones struggling the most), or (2) to use to the students' advantage what they know and are thinking about, and to respond with a set of problems that provide an alternate path to the results, and (3) to modify the difficulty of the problems as the students may be more or less advanced than anticipated and more or fewer lemmas are needed between the main results to keep the majority of the class moving in a positive direction. 

I hope this captures at least some of the key ingredients in the recipe for a successful IBL course. Let me know your thoughts.

The instructor's role in an IBL class, Part 1

In discussing inquiry-based learning (IBL) with college faculty and K-12 teachers, I find that one of the difficult things to do is to communicate what the instructor’s role is, as opposed to what it is not. Many people are familiar with such mantras as, Teaching is not telling, or, Don’t lecture. These are helpful, but then instructors are left wondering what to do. In this post, I want to briefly describe a few important duties of an instructor in an IBL classroom. The roles I am going to describe are not mutually exclusive categories, but interwoven threads. Nonetheless, I call these out because I think they capture some critical aspects of the flavor of teaching an IBL course. These duties are: managing expectations, managing emotions, keeping the students engaged, finding out what students know, and fitting the problems to the students. In this post, I will deal with the first three aspects, and deal with the last two in my next post.

Managing expectations is a primary duty in an IBL classroom. Students come to class, and especially, come to math class, with expectations, including unconscious ones, about what is going to happen. These expectations are often something like, "The teacher will show me a formula and examples, and I just have to memorize and repeat what the teacher does on similar examples." In contrast, in an IBL classroom, students are expected to bring their ideas to problems for which the path to solution may not be clear. Students are not used to being asked to think things through for themselves in math class, and this leads to frustration. The teacher’s first duty is to make it clear to students that they will need to bring their own ideas, and that they will often not know what to do, or they will do things that turn out not to work, but as a class, they will make progress in understanding the mathematics. In class, the IBL instructor can say things like, “This is going to be different, but you will learn a lot,” or, “You’re going to experience mathematics the way that mathematicians do,” or, “You will get stuck a lot in this class. That’s ok. You can even write ‘STUCK!’ on your work when that happens. The important thing is to learn from what you try, both what works and what doesn’t."

In tandem with managing expectations is managing emotions. As mathematicians, we experience frustration as we search for a solution, and we take wrong turns, or the path to the solution is longer than we hoped. Students feel this frustration. If you are managing expectations properly, then students should know that frustration is normal and expected. However, there is more to the instructor’s role than that. If the entire class is boiling over with frustration, the instructor has a duty to respond. If the students are left to flounder, a mutiny can begin to brew. The instructor may say things like, “It seems like this problem/theorem is really stumping us. Let’s brainstorm how we can find new ways to attack it,” or, “I am glad to see everyone is showing persistence on this problem. Sometimes the best way to get past a roadblock is to go around it. So why don’t we look at this {example, related theorem, special case} for now and then come back to the main problem,” or, “This problem is really giving us a rough ride. Let me tell you a quick story about this time when I was frustrated and how I got through it…”

Keeping the students engaged is a multifaceted task. I have written on this blog before about a specific engagement strategy, and about how to reach out to students who are reluctant to participate. I will add to what I’ve written before by mentioning the importance of managing classroom participation. When there is a student presenting work to the class, the instructor’s role is to ensure that the rest of the class is engaging with the presenter and/or the presenter’s work. This can be done in several ways. One of them is to give the students question/comment stems and to call on students to use the stem to offer a question or comment. Some of my stems include, “I like how you…” “What led you to think of…” “I’m not sure I follow how you got from … to …” Another way is to have the presenter pause and to do a check for understanding or a think-pair-share. When the presentation is done, the instructor then pushes the students to ensure that they agree with all aspects of the mathematics, and also see to it that they understand insofar as possible how the student came up with that solution. When students are working in pairs or small groups, the instructor should make sure that all group members are contributing to the conversation. When necessary, this can be actively managed when the instructor inserts him or herself into a group conversation to make statements such as, “I see that Ann and Bob are sharing their ideas. Carol, is what they’re saying making sense to you?” or, “I appreciate that you’re all giving Darryl your attention. Could someone else try to restate what Darryl has shared so far?” or, “It seems like this group has spent some time working independently, each person with his or her own ideas. I think now would be a good time to share some of your progress and see what you can learn from each other’s approaches to this problem."

In the next post, I will take up the other two aspects of the instructor's role.

Let me know your thoughts or other aspects of the instructor's role I have left out.

Formative Assessment

An idea that has been coming up a lot in several different contexts for me is formative assessment. Let me start by stating what formative assessment is, and what it is not. Formative assessment is information gathered from students to assess their current understanding, with the purpose of using that information to make instructional decisions. Formative assessment need not be a formal exam, and cannot be an exam if it will not impact instruction after the exam. Thus, when I am speaking with folks in school districts, sometimes I say “formative assessment” and this is interpreted to mean something like quarterly (or whatever frequency) benchmark tests. In fact, my experience is that teachers rarely are given the time or resources to use benchmark tests as formative assessment. Instead of informing further instruction, they disrupt instruction, as teachers interrupt their regular lessons to review for the benchmark exam, and after the exam, hurry to move on to whatever is next on the overstuffed curriculum pacing guide. As described here, benchmark tests are NOT formative assessment. Instead, when I think of formative assessment, I think of day-to-day tasks that allow the teacher to gather information about what students know, and give the teacher the chance to address gaps and other issues students are having in understanding the ideas of the course.

This semester, one of the best things I learned was how to build questions that would serve as good formative assessment. The questions I have been using were described in my earlier post discussing how I deal with misconceptions. Since this post is about formative assessment, I want to describe how I react to students’ responses to the questions. The questions I have been using are frequently true-false questions, or sometimes multiple choice. Students are first given time to respond to the questions alone (most commonly I have been using Google Forms to collect their initial response), and then to discuss their responses with their peers. Since the discussion may alter their opinions, I then ask for a show of hands for each answer choice. I have seen a few things happen. 

1. Sometimes there seems to be broad consensus on the correct answer. In that case, I will ask one or two students to summarize the reasons for the correct choice, record the answer for the class, and move on.
2. Sometimes, the hand votes are close to equally split between two choices. In this case, I try to get at least one person on each side to articulate the reasons for their answer choice. Then I either ask follow-up questions or I ask other students to add to the arguments for each side. Sometimes this is enough for students to see which is the correct choice. I hear students saying things like, “Oh, I didn’t think of that example,” or, “I changed my mind.” If I feel that there is consensus, then I will record the correct answer at the board and summarize the discussion. If the discussion is not progressing, then I usually prompt students to come up with one or more examples or to draw a graph or diagram related to the statement. Since the topics in these questions are not new, students generally have enough knowledge to resolve the questions. Lastly, I may refer them back to previous work that we did, or pull up the work of a student from the previous class meeting. One of these moves is generally enough to push the discussion toward the correct answer.
3. The third thing that sometimes happens is that I see very few hand votes. I generally take this as a sign of confusion. In that case, I will do one of two things. Sometimes, I tell the students that I see very few votes, and that they need to go back to the discussion with their partner for a couple more minutes to settle things before we can have a class discussion. Then we vote again. More typically, I call on students who raised their hand to explain their thinking, and then call on those who did not vote to see if they are following the argument. After enough students have participated, and I am satisfied that the main ideas have been discussed and reiterated sufficiently, I will ask students for any final questions, and then summarize the discussion and record our answer.  

Formative assessment can be a powerful tool. How do you assess students and use that to inform your instructional moves?

Engage!

The research we have suggests that deep engagement in rich mathematical tasks and student collaboration are keys to promoting learning in mathematics classrooms. For most of the semester, I have been using student presentations to the whole class as the means to engagement and collaboration. However, I have found that it sometimes becomes necessary to adjust the classroom organization. One reason is that running with just one format can become stale over the course of the semester. Also, certain students become comfortable sitting passively in presentations, even though I try to keep them involved. I also find that as the instructor, I sometimes get weary of one routine. In addition, students’ energy levels wane as their workload increases with the end of the semester looming. 

So, I have been mixing it up with a technique I learned as Expert-Home Groups, but it is essentially a Jigsaw. Here’s how I use it:

I assign 2, 3, or 4 problems to the class. For purposes of this example, let us assume two problems have been assigned, #1 and #2.

Divide the room into groups. The number of members in each group is not too important, but usually 4 is the maximum number in a group if they are all going to contribute productively to a discussion. For purposes of illustration, let’s say there are 8 groups, which I will label 1, 2, 3, 4, 5, 6, 7, and 8. Further assume that each group has 4 people.

Within each group, members are assigned a color. With two problems, two colors are needed. Let's say the colors are red and blue. (With three problems, three colors are needed. With four problems, two or four colors can work.) The room then looks like Figure 1, where each digit represents a student.

With the assignments to groups and colors, I now tell the odd-numbered groups to work on problem #1, and the even-numbered groups to work on #2. I tell them that they must make sure everyone in the group understands the problem well enough to explain it to another group.

After the students have had time for discussion, it is time to regroup. Now the red tagged members of odd-numbered groups are paired with red tagged members of even groups, and similarly the blue tagged members of odd-numbered groups are paired with blue tagged members of even-numbered groups, and the room looks like Figure 2.

In the new groups, each problem is explained in turn, until everyone is satisfied that the problems are solved.

Tips for making this work:

• Problems of highly uneven difficulty may not work well. This is because one group will bog down in the problem while another group is idling and waiting to be regrouped. This can be partially counteracted by having groups that finish quickly discuss problems assigned to other groups.
• With two problems and groups of four people, there is slightly less accountability than if four problems and four colors are used. A countermeasure is to use four colors anyway, and regroup the room as in Figure 3. In this way, although there are still two people in each group responsible for the same problem, they worked independently, and therefore may have different solutions or explanations.
• One reason for using two problems rather than four is to deal with uneven numbers of students. If there are some groups with three people, they can still swap a member with another group and thus participate.

Assessment:

• There is assessment, and there is grading. Informal, formative assessment can be gathered during class by listening to the conversations of the groups, popping in to groups with questions, and asking individual group members to respond to questions, to ensure that all group members are participating in and understanding the conversation.
• I have graded these sorts of activities in a few ways. Sometimes I give a participation grade to everyone who appears to be engaged with their group.
• Another way to grade this assignment is to pre-assign each group to its problem (e.g. assign the problem Monday and have the Jigsaw on Wednesday), and then to ask that the group share a copy of their solution with you at the beginning of class. In this way, you have a record that the group (or at least someone in the group) has produced a solution to share.
• You can collect the notes that students take from the class and grade that in a couple of ways. Either it can be used to grade only a group's own problem (so Group 1 is graded for its work on #1, Group 2 is graded for its work on #2, etc.), or it can be used to assess whether everyone is taking notes for all problems as they are sharing.

This is a great way to liven up the classroom and use a little movement to shake things up. If you have opinions on this, or other ways to get small groups actively engaged, I'd love to hear from you.

Dealing with Misconceptions, Part 2

In my last post, I dealt with ways of handling student misconceptions in the moment. In this post, I will discuss how I follow through to make the course better, both from class to class and from semester to semester. I will draw on my Transition to Proof course as the primary example.

I am teaching Transition to Proof for about the fifth time this semester. At this point, I have a good idea of which proofs will cause students the most trouble, and I have specific ideas of what attempts I am likely to see. I have built up this mental cache of ideas by noting what sorts of proofs I have gotten in the past, keeping track of the activities and problems I have used with students, and remembering what kind of effect those items had on the students.

I use this information from class to class to make decisions about whether a misconception from one class needs to be dealt with in the next class. For instance, students had some misconceptions and general confusion around the logical terms "contrapositive," "converse," and "negation." One way I deal with misconceptions is to have multiple problems that center on the same topic, so that we see the same idea come up repeatedly in different ways. In addition, for the first time, I am trying to make use of this information by creating concept questions and using interactive engagement alongside IBL. 

Interactive engagement (IE) has been around for some time, and is probably best established in physics as a mode of instruction that produces significant gains in students' conceptual understanding. More recently, some evidence has emerged that IE has a significant impact in students' understanding of calculus as well (Epstein).

The concept questions I have created are short, multiple-choice or true-false questions that attempt to elicit students' misconceptions, so as to create a space for dialogue that leads students to confront the error in their thinking, and therefore come away with a more robust understanding. In the case of the logical terms, I created a few items that asked questions like, "Which of the following statements is true exactly when the statement, If A, then B, is true?" Or, "Which statement has the opposite truth value to, If A, then B?"

I have been using the questions by projecting them at the front of the class (and making them available for students to view as a Blackboard quiz or Google Form), asking students to answer each question, and then discuss their answers with a partner. Then we go through solutions as a class, usually by having a student explain their answer verbally, but sometimes we may draw a diagram to assist in the explanation. This has led to some good discussions, and has allowed us to zero in on specific issues students are having, without having to prove another theorem or proposition. It's relatively quick and focused on the issues students are having. 

In a recent class, there was disagreement about which choice was an equivalent expression of the definition of one-to-one. Because there were a number of students holding each opinion, there was a lively discussion among students, in pairs or small groups, attempting to decide which option was correct. The room was abuzz with mathematical discussion. As a whole class, I called on students and found three different answer choices that students thought might be correct. Eventually, a couple of students were able to remind the class of what we have learned about the contrapositive, and thereby convince the class of the correct option.

These kinds of discussions are exactly the kind of interactions that let students overcome misconceptions and solidify their understanding of key ideas from the course.

How else do you follow up to address student learning issues?

The Calculus of (Instructional) Variation

As a professor, I take a lot of professional pride in my teaching. As part of that professionalism, I am always looking for ways to improve the learning experience for my students. In this post I am going to describe how some small changes have made a really noticeable impact in one of my classes. (That's where this post title comes from: a little variation has added up to a big change.)

Before I can describe what I did, I should give a little background about what happens in my classes. For now, I am going to focus on my Transition to Proof course. I teach via inquiry-based learning (IBL). As part of that approach, I use student presenters a lot. This means that students come to class having worked on problems (mostly proofs) at home, and they come to class knowing that for most of the problems, someone will have an opportunity to present their proof attempt in front of the class.

This semester, one of my goals is to improve the quality of the discussions that follow a student's presentation. To achieve this goal, I made a couple of changes. In the past, I collected work from everyone at the beginning of class. Then, a presentation proceeded through the following steps:

1. A student wrote their work on the board 
2. The student explained their work.
3. The class proceeded through a Think-Pair-Share: They were asked to look at the work in silence, then share ideas with a partner, and finally ask questions or make comments to the presenter. 
4. During this entire time, the presenter remained standing to answer questions about their work. 

This term, I started in a small classroom with a small chalkboard and a projector screen fixed in place in front of the chalkboard, so that using (most of) the chalkboard was only possible if I unhooked the screen from the wall and set it on the floor. This was part of the inspiration for a new presentation procedure:

1. I photograph student work and upload it to NotesPlus, and project the student work via iPad. 
2. The student explains their work, but sits down immediately, rather than waiting for questions. 
3. The class proceeds through a Think-Pair-Share: They look at the work in silence, then share ideas with a partner, and finally ask questions or make comments, BUT now the presenter is not on the spot during the discussion, as he or she is sitting down.

In addition, I have changed from collecting work at the beginning of class to collecting at the end of class. During class, students used colored pens (Thanks Clark Dollard and Dana Ernst!) to annotate their work, so that I know what was completed before class. Therefore, students are able to compare their work to the work being presented.

These changes are minor, just changing the medium of the presentation, letting the presenter sit during Q&A, and letting students keep their work in front of them for comparison. But the discussions have been stronger for the four weeks of this semester than in years past. My hypothesis is that the students feel more comfortable asking questions with me at the front, even though I am still directing questions back to the class or to the presenter. The class no longer feels like it is putting the presenter on the spot when they raise issues.  Moreover, they are able to ask questions based on their own efforts that they now have in front of them. I am sure there are other factors involved in the improved discussions, including the fact that cohorts of students vary, and this group seems to have a number of people willing to share. Still, it is amazing how small changes can have such a visible impact.