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?

Dealing with misconceptions, Part 1: Seven ways to handle misconceptions in the moment

One reason that I like using inquiry-based learning is that I get to see students' thinking as we progress through the course. When I encounter students exhibiting some kind of misconception, part of me reacts in the moment to find ways to help students confront the misconception, and part of me is taking mental notes about the difficulties that students are having, so as to make use of them in the future. In this post, I take up the first part, the work that I do in the moment, as the thinking emerges.

Let me define misconception. For the purposes of this post, a misconception is any error that I judge to signify a lack of understanding of some aspect of the subject under discussion. I do not intend to count as a misconception things like copying errors, or simple algebra errors that do not change the meaning of the solution to a problem.

In the moment, there are many ways that I might deal with a misconception, and these are somewhat dependent on the misconception. The strategies below are written from the perspective of teaching proof-based class, but apply more widely than that. To wit, here are some strategies:

1. Open a space for peer review. In a group or whole class setting, it is my standing practice not to be the first to evaluate what a student says or writes. So if I see something amiss, I may turn the work over to the group or the class and just open a discussion with, "What do you think?" or "What comments or questions do you have?"
2. Refer students to a definition. One of the core practices of mathematicians is to use definitions. If I see that a student either applies a result not proven, or otherwise deviates from the definition, I will ask all the students to refer back to the definition, read it, and see how it applies to the issue at hand.
3. Ask for an example. If no one volunteers an example, I will ask students to come up with an example. I often tell them to work things out with an example on the side, so that they can see how what is in a proof makes sense (or not) compared with their example.
4. Walk through line by line. Sometimes, I have the students go through each line, read it, and explain the justification for it. 
5. Apply our understanding of the context. For an introduction to proof course, the context may be our concept image (i.e. our prior experience with the concept), for instance, having an idea of what "even" or "function" is supposed to mean. It also refers to applying our understanding of quantifiers. I may ask, "Is this a 'for all' or 'there exists' proof?" "Is this proof covering all possibilities, or is this one special case? Do we need to consider other cases?"
6. Ask a leading question. Here I may focus students directly on the issue. Have we proved this assertion here? Are we sure this is true?  
7. Provide an example to consider. If students are unable to come up with their own example of the statement being considered, I may provide them with an example and ask them to work it through side-by-side with the proof.

Note the move from general, open-ended questions to more narrow, pointed questions or tasks.

After dealing with the misconception on the spot, I try to make a note of what happened. My aim is to capture enough of the misconception so that I can make a decision about how to handle it in a future version of the course. That process will be covered in the next post.

What other strategies do you have? What did I miss? 

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. 

A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins

People who follow my posts to G+ and my tweets may have noticed that I am a reader of Grant Wiggins' blog. Not long ago, he had a post, What is a course? I thought it would be fun to play along. With that in mind, I picked my Transition to Higher Mathematics/Introduction to Proof course. Here are Wiggins' prompts and my responses:

By the end of Transition to Higher Mathematics, students should be able to write proofs and grasp the role of proof as a formal mathematical explanation.

The course builds toward having students prove more logically complex statements and gaining facility with different kinds of proof. The recurring big ideas surround how to attack a proof. We go into depth on key tools like using examples, applying the forward-backward method, using the logical structure of the statement, using definitions.

All of the chapters support these main goals. Students are first introduced to the idea of proof through familiar ideas of number theory and divisibility. They then gain some initial background in logic, and apply it to some number theory proofs before moving on to sets. They are then asked to apply set ideas to sets of real numbers. Then they move to the critical mathematical idea of a function and write proofs about functions and their properties. Finally they are briefly introduced to equivalence relations before moving on to looking at other techniques of proof, and applying these ideas to concepts already seen in class.

Given my priority goals, assessments need to determine whether students are able to demonstrate an understanding of the key approaches to proof, the structure of a logical argument, and to explain the key mathematical concepts of set, open set, function, and properties of a function.

Given my goals, I have exercises and follow-up tasks that should help students gain insight into how the main ideas in key proofs are put together.

If I have been successful, students will be able to transfer their learning to upper division mathematics courses that follow, by attacking proofs with confidence and awareness of the tools available to them, and will persist through difficult courses in the major. If I have been successful, students should avoid such common issues as waiting for the professor to tell them what to do, believing that only others can write proofs, and being uncertain of the role of examples in generating proof ideas (vs. using examples as proof). 

I enjoyed completing this exercise for this course. The course is built as a coherent whole, telling the story of proof and its role in mathematics. To make this work, I have done some tinkering with the emphasis areas of the course, reducing time spent on truth tables (which have a role, but it need not be multiple weeks of a course), and trying not to spend too much time proving things that are too basic. 

What about your courses? Does anyone want to take this up with one of their math courses?

Putting technology to work in my classes

After a summer in which I taught a 3-week workshop for middle school math teachers with iPads in everyone's hands, I am returning to my regular classes, in which students may or may not have a mobile device. What to do? How can I use technology to improve the workflow and the learning in my classroom?

I'm sure there's no one right answer to these questions, so I plan to try 3 variations in my 3 classes. 

For the past several years I used technology mostly outside of class, to create and push PDF handouts to students and to keep a grade book. Having used iPads during the summer, though, I feel that I can improve the workflow and my communication with students by doing more with technology.

For my undergraduate classes:

What am I doing? 

Here, my plan is to take snapshots of students' work to be presented, and to use ThreeRing to manage the photos. The rest of the work (homework submissions, exams) will be handled via paper, though students who miss class can submit homework via emailed photos.

Why?

I am hopeful that snapping photos of the work will free some of my in-class attention to monitor the class better and generally to engage a bit more in-the-moment of the presentation. 

In one of the classes, I am projecting the photo via iPad, and making annotations based on the class discussion using NotesPlus, so that the presenter gets back a photo with the annotations the class made to his/her proof.

In addition, for my undergraduate Math for Middle School Teachers course:

What am I doing?

Here, my focus is going to be on using technology to enliven the curriculum. So I intend to "3 act" some of the material, in the sense of Dan Meyer. I'm working on trying to motivate more of the problems by developing more wonder or want-to-know, using photos and videos as appropriate.

Why?

First, with the onset of Common Core, as much as I think I had a good curriculum, I would like to add another layer of making the problems more intriguing, as opposed to, "Do this because I am assigning it." Also, I want to get these pre-service and early career teachers thinking about how to make lessons that have 21st century appeal, that use media to draw students in to the mathematics. If the problems were not valuable, window dressing would not help; but in this case, I think we had good contextual problems, and adding media should draw out more interest and perhaps help students take more ownership of the directions of the questions we pursue in class.

For a graduate course: 

What am I doing?

Here, the students are primarily practicing secondary teachers, and already have one or more mobile devices (smart phone, tablet, laptop). So I am going to use ThreeRing here also, but adding the feature of having students submit homework to ThreeRing on their own. They also have occasional reading assignments which they submit to a Moodle discussion board (something I was already doing in the past).

Why?

As with the other courses, I want to be freed up from my usual note-taking, and just add a few annotations during presentations, so that I can engage myself more with monitoring understanding and pushing the conversation in the room. Beyond that, I want teachers to begin to see the power of tech tools for rethinking their own classroom workflows.

What did I choose not to do?

I considered:

• Opening a backchannel for students to air their questions during student presentations, using something like TodaysMeet. I have split feelings on this issue. This is either harnessing the power of texting for good, or it is letting face-to-face conversation go the way of the dodo bird.
• Using Subtext, which is going to have a web version soon, to have discussions of readings in grad classes. I have not done this, but will consider it if the web app becomes available, or if we require iPads for our grad students, which we are considering.
• Having students create portfolios in a cloud drive for me to access and give feedback. Here, I decided against it because of access issues. I am surveying my students this semester to see if they have their own devices. But if they only use a school computer, then it is more difficult for them to see my feedback than it is when I return things by hand. Because of the importance of the portfolio as a review tool, I choose not to take that risk here (whereas I do not mind doing this with the presentations being returned via ThreeRing, because they present only a few times, and it is less important that they access the feedback quickly).

What am I missing? How do you feel about this plan? I look forward to your thoughts.

9 Ways to Engage Reluctant Students, aka Tackling the Startup Problem

Any IBL instructor has faced the issue of students who struggle to get going, who seem to want to be passive, and who participate, if at all, only reluctantly. Some years ago, Stan Yoshinobu and I coined "the startup problem" for those students who can't seem to get started with IBL. Let me say up front that I have not solved the startup problem. However, I have developed a number of strategies to use with students who seem disinclined to engage with the class. Let me list them first, and then I will go into detail below.

1. Engage the student in social conversation. 
2. Build habits: set action triggers.
3. Talk about how to succeed in class.
4. Express confidence in IBL.
5. Assign a sociable partner.
6. Invite contributions to discussions.
7. Invite the student to present.
8. Shrink the change.
9. Don't give up.

1. Engage the student in social conversation. One of my first strategies for getting students involved is to make a human connection with them. Ask how they are doing, ask what other classes they are taking, ask about sports or hobbies, but do something so they feel that I recognize their presence in the classroom, and to let them know they are not invisible in the class.
2. Build habits: set action triggers. Sometimes one of the root causes of students' lack of participation is lack of (effective) preparation for class. So I often ask students who are not actively contributing what they are doing outside of class. Based on the information I get, I talk with the student about setting aside a time for working on my class. An action trigger is an idea I learned from Switch. It means you agree to take an action based on some event. In conversations with students, this often sounds like, "OK, so you finish putting your kids to bed, and then that's the best time for you to set an hour to work on this class," or "So, you walk out of the class after mine, and you have a break between classes when you can hide out in the library and work on this class." The idea is just that the student has something external to him or her that will trigger some work time.
3. Talk about how to succeed in class. Closely linked to getting students to make a habit of spending time on the class is helping them understand what to do with that time. In my IBL classes, students are not able to simply mimic a solution provided to them on 20 examples. So we talk about what to do. Read definitions, then reread them. Then try to paraphrase the definition, build examples of the definition, and get comfortable with it. Then read the problem or theorem, and try to understand it with examples. And so on. To the extent that IBL is about not being the source of validation for answers, it is about helping students learn how to learn, which means these kinds of conversations are important.
4. Express confidence in IBL. At this point in my career, I have a lot of success stories, stories of students who struggled at first, but through determination and good work habits, got through my class and went on to graduate. On the first day of class, I let students know that the road will be difficult but worth it. After that, depending on the class and the number of students having the startup problem, either the whole class will hear one of the success stories, or individual students will hear it. I want them to know that I know they can do it, that IBL works, and that I know it will work for them if they put in the effort.
5. Assign a sociable partner. If I think part of the student's lack of engagement is shyness, then I may assign a partner who is confident and sociable, who will engage the quiet student in conversation and encourage him/her to participate in discussions.
6. Invite contributions to discussions. When leading discussions, sometimes I let volunteers share, and sometimes I call on specific students. If I have students who have not contributed in any recent class, I will do a Think-Pair-Share, and then ask a reticent student to share what s/he discussed. By asking for a report on a discussion, rather than asking the student to come up with something on the spot, it relieves some of the pressure on the student, and makes it more likely that s/he will have something to share.
7. Invite the student to present. In my IBL classes, presentations at the board play a role--sometimes larger, sometimes smaller. But in all classes, students are required to present during the semester. If I have students who have not presented, I will seek them out. At first, I may give a non-specific suggestion, like, "I'd like to see you come with a presentation ready in the next week. Let me know." If a week goes by and I get nothing, then I will typically assign the student to a presentation, and remind them that if they are stuck, or worried about it, that they should come talk to me about their ideas in office hours.
8. Shrink the change. Sometimes, even after doing the above, even if the student makes a presentation, s/he doesn't seem to be making progress in the class. If the student is not engaged but attending class, then I will try to find a moment at the end of class when I can catch the student on his/her way out the door, and start a conversation. If the student is not attending class, then they get an email from me. This is where Switch comes in again. I want to get the student emotionally connected with wanting to succeed. Have you ever had one of those "Buy 10, Get 1 Free" cards? Research suggests that people do better with a "Buy 10…" card when they get 2 bonus punches when they start the card, rather than if they had a "Buy 8…" card with no bonus punches. Mathematically, both cards effectively require 8 purchases to earn a free item. But emotionally, we feel farther along with those 3 punches on the first purchase. The term "Shrink the Change" refers to trying to make people feel as if they are farther along than they realize. I will do this both in terms of how far they are toward graduation, and how far they are towards passing the class. Even disengaged students have turned in work, taken the quiz and an exam, etc. So I tell them that they have come this far, and this is what they need to do to complete the course. I also try not to overwhelm them, and I usually focus them on the nearest goal. I might say or write in an email something like, "The first step is to come to the next class with a complete set of attempts on all the problems," or, "The next step is to try this specific problem and either bring a solution to class or bring your ideas to me before class, so you can present a solution."
9. Don't give up. I have been using IBL for more than 10 years. I have seen some surprising cases where students who did not seem to be making progress, even 10 weeks into a 15 week semester, somehow found their way to success in the class. I remind myself of this fact whenever I encounter tough cases. Sometimes if I show faith in the students, that is the small push they need to find the determination to succeed. 

Harnessing your personality

This summer, I had the privilege of working with some great people, including my longtime friend and IBL (Inquiry-Based Learning) blogger Stan Yoshinobu, Dana Ernst, Dylan Retsek, and a number of other IBL instructors. One day, we sat on a panel and we were discussing our approaches to getting student buy-in in our courses. As we went around, I was struck by the variety of ways we had to approach this issue. Dylan Retsek described being William Wallace, rallying students into a frenzy of excitement, and Dana Ernst added to that by saying he tries for Robin Williams, i.e., using humor, and William Wallace, and mentioned being a cheerleader for students. I think both Dylan and Dana are excellent instructors, but I doubt I could run my class that way. 

I do use some humor. On the other hand, I doubt that I have ever led a Wallace-like rally, and I don't think I would describe myself as cheerleading either. My classroom demeanor is very low key. In some classes, I have made a straight face and said, "This is what it looks like when I'm not excited about your work," followed by making the same face and saying, "This is what it looks like when I AM excited about your work." In classes where there is a lot of math-phobia, I set the tone for the course early on by having them share how they feel about math. Then I note how many negative attitudes toward math there are, and I tell the class that I want to help move them in a positive direction, but that we will need to do things differently. 

Instead of cheer, I usually offer a simple thank-you to a student who presents work to the class, and I try to include specific points of recognition and suggestions for improvement.

The point is that everyone has their own personality, and making IBL work in YOUR class will require that you find the elements of your personality that help you identify with students in their struggle to learn, and that assuage their fears. Sometimes, instructors let themselves out of implementing key portions of IBL because it they don't feel it fits. I am suggesting that certain kinds of actions are critical, but that you have to find the way that makes them feel right to you.

IBL instructors have to connect with students, and work to communicate to students that the IBL approach will work, but will take patience. This could be rallying them with excitement, or helping them find the feelings that suggest something different is warranted. IBL instructors tend to value giving recognition to students who share their ideas, but the way this is delivered will be based on your personality. 

To borrow an analogy I sometimes use in class, students in an IBL class need to know that as in a swim class, they must be the ones doing the swimming--you can't do it for them--but that you will not let them drown. HOW you communicate this sentiment is up to you.

Make IBL work in your class by finding ways to harness your personality to deliver to students not only your high expectations, but also the message that you will help them find success.

App usage in a mathematics institute with iPads for teacher participants

I recently hosted a 3-week institute for middle school mathematics teachers in which all of us were using iPads, and we operated in a (near) paperless environment. As I prepared for this, I was struck by how often I ran across lists of apps, and how rarely I read in-depth discussion of how to integrate various apps into a real classroom. Maybe I just wasn't looking in the right places. Nonetheless, this is my small contribution to providing a guide to using apps to support discussion and learning, which I hope will be informative for anyone else, especially in mathematics, looking to make the leap to a one-to-one iPad environment.

Introduction of the iPad:

Of the teachers who signed up for the institute, about two-thirds were new, whereas one-third had already received an iPad last summer and were back to learn more. New participants were introduced to the iPad, shown a few multi-touch gestures, and asked to download a number of apps on a day the week before the institute. This was important because they were unable to complete downloads over the course of an entire morning (too much bandwidth usage in one room).

Sharing content with teacher attendees in a paperless environment:

I worked with a team of facilitators. I created a shared folder for content on Dropbox, so that all of the facilitators could access and add materials. 

I created a master schedule as a Google spreadsheet with hyperlinks to some of the virtual handouts for teachers. The main reason for using a Google spreadsheet was that the hyperlinks in an Excel file saved in Dropbox, for instance, are not active if viewed in Dropbox. Moreover, I did not want to save as a PDF with hyperlinks because I wanted to be able to revise the schedule, and have the teachers' schedule auto-sync with the revisions. I should have told everyone (but only told a few) that they should download Google Drive or bookmark the site for the schedule. In the absence of downloading or bookmarking, teachers had to re-locate the email invitation to view the schedule again.

Note that a shared folder on Dropbox enables those sharing the folder full editing privileges, including the possibility of deleting documents. For this reason, and also because it is sometimes a hassle to search through a folder for the one document needed at a particular moment, teachers were not invited to share the folder with facilitators.

Instead, teachers had already downloaded the Lino app. I sent them a link to a Lino board. They were asked to copy that link onto their own Main Lino board, to save them from having to find the email over and over. (Note that inviting someone to a Lino board does not add that board to "My Boards".) Facilitators then put individual file links on the Lino board, and the file links would lead to particular items in the facilitators' shared Dropbox folder. In this way, participants could download content from Dropbox and keep the content wherever they wished, typically in their own Dropbox, in iBooks, in NotesPlus, or in PDF Notes. 

Doing mathematical work:

When working on mathematics, the problem handouts were almost always created as a PDF and a Dropbox file link shared on the Lino board. For the mathematical work of the teachers, the lead facilitator would create a Baiboard (I think the pronunciation is Bye-board, because the Mandarin for "white" is "bai," pronounced like the English word "bye") and share the board number by writing it on a physical whiteboard in the classroom. The facilitator then asked teachers to post a screenshot of their work (usually done in NotesPlus, perhaps with the help of the TI-Nspire) on a particular one of the 6 pages in a given Baiboard session. Besides being limited to 6 pages, the other limitations of Baiboard are that there is a slight delay between actions being taken by one user and being seen by the other users--however, this delay is not long enough to cause problems. Also, sometimes other teachers would log in to the same board and occasionally, inadvertently, move the photo on a board. 

Note that teachers shared screenshots this way, and not via Lino, because photos posted directly to a Lino board are too small for viewing details. Using Baiboard was also shorter than having teachers export their NotesPlus work as a PDF, saving it to Dropbox, copying the link, and sending the link or posting it in Lino. We had also considered using Instashare for participants' screenshots, but Instashare was unreliable, as it would claim to be sending, but the recipient would not see anything delivered.

In the previous year, facilitators used Reflector to show teachers' work, by having particular teachers connect their iPad wirelessly to a laptop at the front of the room. This year, we did not use Reflector, because for some reason the laptop we were using had trouble getting the signal from the iPad. (It may be that with 40 iPads in a room, there is too much signal noise.)

Apps for specialized duties, Part 1:

On occasion, facilitators ran a session via Nearpod. Since there were more than 30 teachers, it was not possible to have a session running the free version of Nearpod with everyone logged in. Instead, typically one member of a pair or group was assigned to log in and report responses from their work in partnership or in groups. Nearpod saves the responses from a session, which aids in later analysis. I used one of the sessions and created a word cloud with the responses.

I used SurveyMonkey for end-of-day feedback. I used individual email links, and asked teachers to copy the link from the email onto their own Lino board (either Main or one of their own creation).

I wanted to make new groups of teachers each day, so that teachers would meet people from other neighborhood schools, rather than clustering with the teachers from their own school. I used GroupMaker each day to set up groups. It was good in that it gave me a reason to take photos of teachers and helped me put names to faces much more quickly. I used Create Groups each day to make new groups. Within any group, the "random" assignment to groups appears to work only once. That is, you cannot re-sort teachers randomly into groups. After one week, I had groupings named for each day of the week, and I didn't want to keep creating more, so I manually regrouped the participants.

We also tried The Answer Pad (the student app is TAPit). When I tested this with a small group, I set up the list of users and (simple) passwords myself. When I scaled up to the class, I wanted people to register themselves. I had a task in mind for which I wanted to collect teacher responses. I planned to have teachers register and then immediately respond to the task. Self-registration did not work well for this purpose. I had to keep resending the "test" to the new registrants, and even then, not everyone could get in to see it. If I were using it over an entire semester, I would try again, since self-registration is a one-time set-up for a course, but for a single use with a large group, it did not work out.

We spent some time teaching the basic use of the TI-Nspire CAS app, and revisited it as a tool for mathematical modeling on subsequent days. The TI-Nspire seems to be the best app out there right now for doing secondary-type mathematics. We used it to enter data and create linear and quadratic models, to do visual line-of-best-fit, to run simple calculations, solve equations, and to do a summation.

As an alternative to the TI, Wolfram Alpha is nice, but the single line interface means it is hard to edit commands, once entered. Also, if one wants to compare, say, a linear model and a quadratic model of a data set, Wolfram Alpha is not set up to do that.

Apps for specialized duties, Part 2:

These are some of the apps we used, but not to the extent of ones already mentioned.

We used Socrative on the first day. Socrative is convenient for impromptu queries, like classroom clickers, but it does not retain answers from that form. You have to use quizzes instead if you want to view the data later.

Teachers wrote lesson plans and occasionally were asked to take notes in Pages. Unfortunately, when Pages opened Word tables, the formatting turned into a mess, which took a lot of work to fix. But beyond this, Pages is a nice document editor.

Teachers were asked to create a Keynote about one of the lessons they created in the institute, and then to export the Keynote as a PDF in Dropbox, and finally to upload the PDF into Nearpod, and add a poll, so that their viewers could give them feedback. This entire cycle worked out well. Teachers got the full experience of using Nearpod, including visiting the website afterwards to view the feedback they received. Nearpod keeps records of data (i.e. responses) collected during each session. 

The Common Core app was available as a reference.

We showed participants Educreations and ShowMe. In one session, we asked teachers to record their solutions and share them at the front of the class by physically bringing up the iPad and connecting to the projector, but many were embarrassed about hearing their voices. Perhaps if we had made this a regular feature, the embarrassment would have been alleviated.

We spent two sessions using Algebra Tiles, one for balancing equations, and one using it for understanding integers as hot and cold cubes.

We spent a session with iThoughts as a tool for pre-planning lessons.

We spent a session showing the basics of GeometryPad. It quickly became apparent that many more features can be unlocked in the non-free version.

We did one session incorporating Ubersense for its slow motion and freeze frame features.

We had one session where we gave a brief mention to each of several apps that we were not otherwise using in the institute, but that could be helpful: 

• ClassDojo
• TooNoisy
• MyScript calculator
• Slope Slider
• Sketch2Graph
• Remind101
• ThreeRing

I look forward to hearing from any of you out there that have worked with these or other apps. What was your experience? Are there apps that do some of this work better than the ones we chose? Are there ways of using these apps to make them run better?

What is The Math Switch?

I've named this blog The Math Switch for three main reasons. First, the book Switch, by Chip and Dan Heath, is one of my all-time favorite reads. It highlights a lot of important features that are involved whenever one is trying to effect change. Perhaps most important among these is that appeals to reason are often ineffective by themselves; more often, an emotional appeal is what matters most. As a professor of mathematics, one of the things that I face is that, at least in the US, there is widespread aversion to mathematics; thus, I expend a lot of effort trying to help people switch perspectives, from one that sees mathematics as abstruse, irrelevant, and something only geniuses can do, to a subject about important ideas, relevant (yet also an art form), and something that ordinary people can do if given time, and given the chance to see that mathematics is NOT particularly about computing things with formulas, but about a particular kind of reasoning and a way of working with ideas using precise definitions and logic. I love to turn on the mathematical switch in people, and I love to help other teachers and professors learn how to switch people on to mathematics.

A second reason for The Math Switch is that a switch suggests turning on an electronic gadget. A few years ago, I was highly skeptical of technology as a classroom tool. My previous experience in teaching the basics of some software tools useful in mathematics (Excel, Geometer's SketchPad) led to a lot of frustration and was as much a distraction from the mathematics as it was an aid to learning. However, in 2012 I became a part of a project that set out to give iPads to math teachers, and to help them learn mathematics and learn to teach mathematics using the iPad as a tool. In spite of my doubts, I was determined to find ways to make the iPad a tool for teaching and learning. I've learned a lot since then, and I look forward to learning a lot more. As I learn, I hope to share ideas and to gain from the wisdom of others out there.

The third major reason for the name The Math Switch is that a switch suggests turning on a light. In this case, I want to shine the light on people, as well as tools, both pedagogical and technological, that are making good things happen when it comes to teaching and learning.

Welcome to The Math Switch. Bring your bright ideas.