Curriculum Development Instruments

Student Interviews

Instrument 9: Combinatorics Content Interview Protocol

Project: Teaching Introductory Combinatorics by Guided Group Discovery
Dartmouth College

Funding Source: NSF - Division of Undergraduate Education (DUE)

Purpose: To get student feedback about a course, primarily by gauging in an informal manner what they learned, then eliciting ratings by the interviewer and the interviewee of how the interview went

Administered To: Post-secondary students

Topics Covered:

• Content Specific Assessment: mathematics combinatorics
• Self Assessment (Student): confidence, comfort, understanding, anxiety, metacognition

Format/Length: 12 open-ended items

Combinatorics Content Interview Protocol

(Sample preface.) Thanks for taking the time to do this interview. As you may know, this course is part of a project that is introducing some new features to the Combinatorics course. The purpose of this interview is to get feedback that will be useful to the instructors as they develop the new course. Remember, this is not a test of you, it's a way for us to find out more about how the course works. I hope you'll take your time and give us thoughtful answers. That way we get the best and most accurate understanding of the course. Of course, everything you tell me is confidential and will be reported so that no one's opinions can be identified.

GENERAL QUESTIONS

1. First, tell me a little bit about your math background.


2. Why did you decide to take this course?


3. Looking back on the course as a whole, do you see any main ideas or themes that seem to run through the course?
   [If at this point they mention any of the main concepts queried in the interview—blocks of a partition, inclusion and exclusion, generating functions, the product principle or recurrences—go directly to those questions in the order mentioned by the student. E.g., "You mentioned the product principle, so let's talk a little about that. Can you think...."]

SPECIFIC QUESTIONS

1. "Are you familiar with the idea of a partition of a set? of an integer?"
• Can you give me an example of that?
• Now let me play psychoanalyst and ask you to free associate about the idea of a partition of a set.

Probes:	(1) back up—ask for a definition.
	(2) offer a simple problem, such as computing the number of partitions of a 5-element set into 3 blocks, or distributing five distinct pieces of candy into three identical paper bags.



2. "Are you familiar with the idea of counting the blocks of a partition of a set?
• Can you give me an example of that?
• Now let me play psychoanalyst and ask you to free associate about the idea of counting the blocks of a partition.

Probes:	(1) back up—ask for a definition.
	(2) offer a simple problem, such as seating four people at a table or a simple set question.



3. Do you recall using certain counting principles in the course?

Probe:

For example, the sum principle or the product principle?

• Can you give me an example of where one of these principles is used?
• Can you think of any other principles?

Probe:

Pigeon hole, induction, quotient, bijection



4. Can you tell me what a bijection is and why it's important in combinatorics?


5. Now I'd like to talk about the idea of inclusion and exclusion.
• Can you give me an example of how you would use this to count something?
• [Mike poses a scenario.] Tell me how you would go about setting up this problem.

Probes:	(1) How do you figure out the size of a union of two overlapping sets? three overlapping sets?
	(2) Suppose you are counting arrangements that might or might not have certain properties. How many have none of the properties?



6. Are you familiar with the idea of generating functions?
• [Mikes sets a problem, such as k element subsets of an n element set when n is fixed.] How would attack a problem like this?

Probes:	back up—remind about binomial theorem
	back up—what happens if I take (1 + x)2.



7. Let's talk a little bit about the product principle.
• Can you think of an interesting example of how it's used?


8. I'd like to ask you to think about recurrences.
• Can you explain Pascal's recurrence?
  Did you use any big principles of combinatorics to explain that?
  .....If answer easily, then: Can you give me an example of an "interesting" recurrence?
  .....If answer with difficulty, then ask about "with n or without n."


9. Finally, would you look at these problems and tell me how you would go about setting them up? Just talk me through what's going though your head as you think about how to solve these.
   Mike sets several template problems to see whether students rely on templates for solving or begin from deeper principles.


10. Do you feel that you are better at monitoring your own thinking about mathematics now than you were when you started the course? Why?

(New page)

Faculty Assessor Questionnaire

1 = disagree.........6 = agree

Grade

(New page)

Student Questionnaire

1 = disagree.........6 = agree

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