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.