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.  

Working in the Cloud, Part 2: Google Drive

In my last post, I covered my usage of Dropbox. In this post, I am going to pick up with my usage of Google Drive.

When I only had home and office desktops and a laptop, I was satisfied with Dropbox alone. Two years ago, I got an iPad, and began to explore more apps, including Evernote and Google Drive. Given that I was already a Dropbox user, I have added the other services to fulfill particular needs. If I had come to the others first, my usage of them would probably be different. In this post, I will discuss Google Drive.

Whereas Dropbox is really about file storage, Google Drive is a more expansive system. You can use Google Drive to store files of all types. However, files in Google Drive work especially well with Google’s Documents, Sheets, Forms, Presentations, and Drawings. For me, Google Drive serves to enable collaboration in ways that are difficult with Dropbox. In particular, if I have a document that I am going to co-author, and I want my co-author to be able to view the document with me simultaneously, then I will use Google docs. I also like the feature that allows me to select what people can do with an item I share, where the options are: can view/can comment/can edit. So when I am running a workshop, for instance, my co-facilitators may have editing capability, while participants get viewing capability. And, unlike Dropbox, only files that originate with me count against my file storage limit.

My favorite feature of Google Drive is Google Forms. Google Forms are great for surveys or quizzes. There are several different question formats that you can set up for a question, including multiple choice and short answer. The responses to the form are collected in a Google Sheet. And, with the use of Flubaroo, an add-on to Sheets, I can auto-score a quiz as well, and have the scores emailed to the students.

There are a couple of nice aspects of storage in Google Drive. One is that Google’s native formats (Documents, Sheets, Forms, Presentations, and Drawings) do not count against your storage limit. Another is that items scanned in to Drive get Optical Character Recognition (OCR) applied to them, so that if, for instance, I scan a hard copy of a typed document, I am saved from re-typing it, because OCR converts the scanned content into text.

Google Drive has its own set of limitations. One of the limitations across Google’s native Docs and Forms is that it is difficult to typeset mathematics. (I have a partial work-around, but I’ll save that for another day.) Another issue is that Documents and Sheets are editable on the iPad, but at last check, Drawings and Forms are not. Also, where Dropbox files are stored locally on a laptop or desktop, so that you can work offline, and particular files can be selected for local storage on your tablet, Google Drive files are not generally available offline. I have had the experience of trying to access a file when I have a slow internet connection, and I am stuck unable to access it. (If you use Chrome or Chrome OS, you can set up offline access: https://support.google.com/drive/answer/2375012?hl=en.)

Google Drive Summary:

Advantages:

+Native editor for proprietary file types.

+Native Document and Sheet editors work on the iPad.

+Forms are great for surveys and quizzes. 

+Form results go in a spreadsheet that can be auto-scored with Flubaroo.

+Files in native Google formats do not count against the storage limit.

+Only my own files count against my storage limit, and not those files shared with me but owned by others.

+OCR lets me convert hard copies into electronic text.

Disadvantages:

-Mathematics is difficult to typeset in Google’s native formats.

-Google Forms, Presentations, and Drawings cannot be edited on an iPad.

-Google files are not available offline except via Chrome or Chrome OS.

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.