6 Ways for Students to Grapple with Definitions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Formative Assessment

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

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

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

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

Engage!

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

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

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

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

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

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

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

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

Tips for making this work:

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

Assessment:

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

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

Dealing with Misconceptions, Part 2

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

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

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

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

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

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

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

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

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

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?