teaching portfolio - university of michigandavissch/cv/tp 11-2016.pdf · this teaching portfolio...

Teaching PortfolioDaniel Visscher

November 2016

Table of Contents

Teaching StatementTeaching Experience

and TrainingCourse Snapshot:

Explorations in Topology and Analysis

Course Snapshot: Mathematics for Elementary School Teachers

Evaluations

1

2 3-6

7-9

10-12My class is working on a proof of the Bolzano-Weierstrass Theorem. To date, this is the most challenging proof I have asked them to create. I have scaf-folded the worksheet so that students are inclined to use a set of tools we have previously encountered: bisection, iteration, and nested intervals. Af-ter giving groups a while to think about the problem on their own, I sit down with a group. They have produced a sequence of nested intervals and found a convergent sequence via the Axiom of Completeness, but their method is based on starting with an already convergent sequence. I make a habit of asking about solutions regardless of their correctness, so my next ques-tion will not be a leading one. I ask how they know that exactly one of the intervals resulting from a bisection contains infinitely many terms of the sequence. There is some disagreement. One student produces an example for which their method works. I ask if there are sequences that are bounded but don’t converge, and another student volunteers an example from the previous worksheet. I suggest they check to see if their method also applies to this case and, if so, what convergent sequence it produces. They begin a refined investigation.

- t.o.c -

This Teaching Portfolio documents my teaching philosophy and experience. It contains a compilation of course materials that I have developed, student work, classroom observations, and course evaluations. The Course Snapshots provide a look at the mate-rials and methods I have used in two specific courses. End-term evaluations, as well as a compilation of com-ments from student course feedback and evaluations, can be found in the Evaluations section.

PAGES “...I hope I read about some Visscher theorem or something one day knowing he went off and did something he clearly loves. Again I CANNOT overstate how happy I am with him and this class.”

- Student evaluation (Fall 2015)

“Professor Daniel Visscher is a god among mortals. This class is difficult, but he makes an effort to teach us material and bestow a glorious sense of understanding among us...If the great Imperator Caesar Augustus of Rome were a modern day math teacher, he would bow and give deference to the great Visscher. (In all seriousness, this is a nice section. I was worried about how difficult it would be, but I’ve learned a lot and I’m having a good time with the material.)”


“I loved the seminar atmosphere. The small class size enabled us to work as a group and helped my learning. The grading was hard - especially on the homework though it ended up just pushing us more. My writing in this class improved extensively because of this... Overall I loved this class, learned loads from Dr. Visscher, and will miss this class when we are finished.”

- Student evaluation (Winter 2013)

“The thing that I liked most about this class was that it was never discourag-ing. We never were overwhelmed or overly frustrated when solving prob-lems because the process of finding a solution was valued more than the solution itself. It was fun to be in a class where risks were encouraged and mistakes were valued because they lead to further understanding.”

- Student evaluation (Winter 2014)

“He seemed dedicated to our learning and to the inquiry approach. He refused to ever just supply us with answers and instead would help to guide our thinking through suggestions or questions. His office hours were particularly helpful. Dan is really approachable and easy to talk to one-on-one.”


“I really like the way you structure calc class, and it has become a class I look forward to! I find learning about the way people learn interesting, especially the growth vs. fixed mindsets... [I] am glad that you are helping us learn about the way we learn!”

Midterm student comment (Fall 2015)

The students described the class as a “safe environment for learning.” The students said they are comfortable presenting in class and sharing their ideas with Professor Visscher. The students described Professor Visscher as “very responsive, caring and thoughtful” and said that they felt that “it is OK to be wrong.” The students also appreciated that “there is enough time for group work” so they don’t feel rushed.

- Midterm focus group (Fall 2016)

Teaching Philosophy As a mathematics instructor, I position students so they are able to construct and organize mathematical knowledge. This means helping students to develop the tools they need, as well as giv-ing them opportunities to test out and explain their ideas. I believe this promotes an understanding of mathematics as a living discipline. It posits students’ work in the classroom not only in the service of acquiring mathematical knowledge but also of developing mathematical practices. I do this with four central objectives in mind: I want my students to actively do and communicate mathematics, to think deeply about mathematics, to appreciate and benefit from the variety of ways people think, and to connect the content and skills they learn in my classroom to their broader education.

In my classroom, students actively do and communicate mathematics. That is, students solve problems and prove theorems in class, and they do this in collaboration with their classmates. I facilitate this process by designing problems that steer students to encounter the big ideas and the common misconceptions around the mathematics at hand. My goal is to set students up for productive conversations. They often work in teams on common solutions at chalkboards around the room, which physically engages them in doing mathematics. This requires students to com-municate their ideas and make their thinking visible. Afterward, we talk as a class about the big ideas, challenges, and breakthroughs that groups encountered, uncovering the major themes I want students to take from the lesson.

[Dr. Visscher] was really good at guiding us through problems in a way that allowed us to discover the answers for ourselves... I learned a lot through talking through my work and discussing the challenges and successes that I had. —Student evaluation

In addition to in-class activities, I design assessments that engage students in professional-style communication of mathematics outside of the classroom. In my topology and analysis class, stu-dents produce a course textbook by writing proofs of theorems, crafting definitions, and construct-ing examples. In my Euclidean geometry class, I am the managing editor for a professional style journal, Communications in Euclidean Geometry, to which students submit articles for publication and serve as referees. I believe these activities help transform students’ understanding of doing mathematics from a one-time event of computing a solution to a process of refining, clarifying, and more deeply understanding ideas. Those experiences have a positive, lasting influence on students’ beliefs about their mathematical efficacy.

In my classroom, students think deeply about mathematics. One way I promote this is by pushing my students to be arbiters of mathematical correctness. This requires that they fully engage with what goes on in the class. Students know that they are responsible for making sure what is said and written is correct. When the class is hesitant to engage with a proposed solution, I ask questions to spark discussion: Where does this proof use the assumptions? Does this reasoning still work if we change a part of the problem? Which parts of the proposed solution make you feel uncomfortable? I also ask questions of correct answers, so that my voice is not an implicit evaluation of the quality of a solution. This process is additionally important because it provides a vehicle for discussing un-derlying concepts—methods or theorems that were important or problem solving techniques that were useful. By building a culture of thinking about a problem after students solve it, students gain not only solutions but also a deeper and more flexible understanding of their solutions, and they create a skill set for solving future problems.

Thinking deeply about mathematics in my classroom also involves listening intently. This is true of students listening to their peers, and it is true of me listening to students. To learn about my stu-dents and their progress, I sit down with groups and observe them work. I ask students to explain

their thinking, but my objective is not just to hear a certain sequence of words. Students are learn-ing to speak a language, and sometimes what they say is not an accurate representation of what they understand. This can be the case when the spoken words are incorrect or correct. So, I ask lots of questions of my students, even when an initial answer sounds good.

In my classroom, students experience a diversity of backgrounds, learning styles, and ways of thinking, and they use these as assets. Professional mathematicians have different strengths that make them “good at mathematics,” and I believe the mathematics community benefits from this diversity. I foster a similar community for students by creating a collaborative classroom. Stu-dents work in groups that change regularly, exposing them to many di erent styles of thinking dur-ing the problem solving process. This requires students to interact with others who are not similar to them, developing a skill I believe is important in a much broader context than mathematics. I encourage my students to think of what their strengths are and how those can contribute to the mathematics classroom. These activities challenge the perception of mathematical skill as a single, fixed trait, and they expand students’ ideas of who is good at mathematics.

In order for students to benefit from their classmates’ ideas, however, I need them to be willing to take risks sharing their thoughts. One way I promote this is by discussing how mistakes help us to appreciate what is confusing or difficult about an idea, which can bring us to a be er understand-ing of it. A student in my analysis class reflected that “it was fun to be in a class where risks were encouraged and mistakes were valued because they lead to further understanding.” In my class for pre-service elementary teachers, I have students report on their “productive failures” in class. This encourages students with low self-efficacy to participate, and, in turn, the classroom is a more lively and intellectually diverse place.

In my classroom, students connect what they learn to their broader education. Many of the learning goals for my courses are not content specific. In my linear algebra class, for example, I want students to use mathematical language and symbols as a way to clarify and make intuition precise. In my calculus class, I want students to be able to form valid judgments from quantitative informa-tion. I also want students to realize that even intuitive ideas can and should be subjected to rigorous scrutiny. I advertise these objectives at the beginning of the term, and I periodically ask students to reflect on their progress. I also point to these objectives as we encounter them. In analysis, a proof of the Bolzano-Weierstrass Theorem is important in the narrative of the course. It is also a mean-ingful demonstration of the mathematical use of bisection and the Axiom of Completeness, as well as the power of abstraction and making intuition precise. The ability to abstract from a particular case to general principles, articulate thoughts precisely, and critically evaluate the logic of a state-ment are skills that benefit students as they work on a chemistry lab, write an English essay, or listen to the news. I encourage students to reflect on these kinds of implications.

My philosophy of teaching mathematics is closely tied to my own experiences of doing mathe-matics. I find research most enlivening when I am actively exploring, asking questions, and talking with others about mathematical ideas and problems. My moments of insight and progress typically come from understanding one theorem or object very deeply or by comparing examples. I learn new skills and construct better ideas by collaborating with colleagues whose strengths are different than mine. Finally, I find that the logic I use and the questions I ask as a mathematician influence how I interact and engage as an academic, teacher, and citizen. It is my philosophy that involving students in an active learning classroom helps them not only learn and retain mathematical knowl-edge but also gain invaluable skills and experiences through the process of mathematical discovery.- 1 -

* This is part of an inquiry based learn-ing textbook, Explorations in Topology and Analysis, which I am writing with Alejandro Uribe.

** This activity is inspired by T.J. Hitchman’s course “Euclidean Geom-etry: An Introduction to Mathematical Work”.

*** Amie Wilkinson gave a talk in 2013 at the University of Michigan for the Women in Mathematics Club, in which she presented the results of a survey asking mathematicians to evaluate their strengths and weak-nesses among thirty-one characteristics that one might reasonably think would be helpful to a mathematician (thorough, geometric thinker, quick, curious, patient, communicator...). Among seven well-known mathemati-cians, no characteristic was a univer-sal strength and most characteristics were self-identified as a weakness by at least one person in the group.

**** A productive failure is a mistake or misconception that, in the end, leads to a be er understanding of the mathematics. I learned about this technique from Stan Yoshinobu at an MAA PREP workshop. I believe it makes a valuable contribution to the classroom culture by encouraging stu-dents to reflect on how their mistakes can be productive as well as com-bating the common misconception, especially held by students who feel they are struggling, that mathematics is simple for other students.

# This is also supported by research. See, for instance, papers by Sandra Laursen and collaborators at the Uni-versity of Colorado at Boulder for the positive effects of inquiry based learn-ing, especially for underrepresented groups.

*

**

***

****

#

Teaching ExperienceInstructor, University of Michigan-Ann Arbor

I have taught active learning and Inquiry-Based Learning (IBL) courses through the De-partment of Mathematics. Math 116 is part of an active and collaborative learning cal-culus program that has been written about in MAA publications as a successful model. Math 217 doubles as the department’s introduction to proof course. I am co-authoring a textbook based on the course notes for Math 176.

116 Calculus II 176 Explorations in Topology and Analysis (IBL) 217 Linear Algebra (IBL) 385 Mathematics for Elementary School Teachers (IBL) 431 Topics in Geometry for Teachers (IBL)

Instructor, Michigan Math and Science Scholars (UM)Michigan Math and Science Scholars is a summer enrichment program for high school students. I taught a two week intensive course that met every morning for three hours and supervised a graduate student who ran a complementary afternoon session.

MMSS Discrete Dynamical Systems

Instructor, Northwestern University School of Continuing Studies I taught three calculus courses through the School of Continuing Studies. Each of these courses ran independently from other calculus courses, and I was solely responsible for their design and implementation.

220 Differential Calculus of One Variable Functions 234 Multiple Integration and Vector Calculus

Instructor, Northwestern University EXCEL ProgramI taught a Calculus II course as part of a summer program for diverse and underrepre-sented students entering their first year of college in the School of Engineering. The intent of the program was to develop a peer network around academics before starting college in the fall, and to encourage students to assume a leadership role in their courses.

224 Integral Calculus of One Variable Functions

Teaching Assistant, Northwestern UniversityMy responsibilities included leading recitation/quiz/problem sections once a week; hold-ing office hours; grading homework assignments, quizzes, and exams; and occasionally lecturing material.

Game Theory (Math 104), Finite Mathematics (202), Differential Calculus of One Variable Functions (220), Integral Calculus of One Variable Functions (224), Dif-ferential Calculus of Multivariable Functions (230), Linear Algebra (240), Linear Algebra and Multivariable Calculus (290), Foundations of Higher Mathematics (300), Geometry (340), Chaotic Dynamical Systems (354).

Training and Professional Development

Project NExT2015-2016

I participated in the MAA’s national professional development program. This program addresses “all aspects of an academic career: improving the teaching and learning of mathematics, engaging in research and scholarship, finding exciting and interesting ser-vice opportunities, and participating in professional activities.”

Inquiry-Based Learning WorkshopKenyon CollegeJune 2014

I attended an MAA PREP workshop on how to run an Inquiry-Based Learning class-room. We observed and discussed how other instructors implement IBL, and we pre-pared materials for our own IBL courses under the mentorship of experienced IBL in-structors.

Mathematicians in Mathematics Education WorkshopUniversity of ArizonaMarch 2013

I participated in a workshop led by William McCallum (one of the authors of the math-ematics standards for the Common Core) focusing on how mathematicians can become more involved in K-12 mathematics education. The workshop addressed issues such as standards development, curriculum design, and preparation of future teachers.

Ann Arbor Elementary Mathematics Laboratory and WorkshopUniversity of MichiganAugust 2012

I participated in a workshop centered around a laboratory school led by Deborah Ball for rising sixth graders struggling with mathematics in the Ann Arbor and Ypsilanti School Districts. Participants in the workshop included in-service teachers, mathemat-ics education researchers, and course instructors for pre-service teachers. We observed and discussed issues pertaining to both content and methods in teaching elementary mathematics.

Graduate Teaching Fellow, Graduate Teaching Mentor, Graduate Teaching Certificate ProgramSearle Center for Teaching ExcellenceNorthwestern University2009-2012

As a graduate student, I participated and helped to organize programs to support teacher development through the Searle Center for Teaching Excellence. I completed a Teaching Certificate Program in 2009-2010, was a mentor for that program the follow-ing year, and designed and facilitated teaching workshops for graduate students as a Graduate Teaching Fellow in 2011-2012.

Teaching Experience and Training

overview teaching experience

training and professional

development

- 2 -

My teaching record reflects my work with different populations of students, including students in a school of con-tinuing studies, a pre-college program for underrepresented students, an hon-ors college, and a pre-service teaching program.

I have teaching experience with a variety of mathematics courses. I am prepared to teach, in addition to the core mathematics courses, math for non-majors; teacher preparation courses; and upper-level courses and independent studies in analysis, geom-etry, topology, and dynamical systems. I am also interested in teaching courses in abstract algebra, combina-torics, and graph theory, having previ-ously conducted research relevant to these areas. Furthermore, I have enjoyed developing my knowledge of mathematics education while teaching Math 385 at Michigan, and I would be excited to expand my knowledge and interests in statistics or modeling.

I have been active in teacher train-ing and professional development, both as a participant and as a mentor. I think it is important to continually learn new things about teaching and learning, as well as to share my own experiences with others.

Course Information Math 176: Explorations in Topology and Analysis

Instructor Daniel Visscher ([email protected])3827 East Hall, Office Hours: TTh 1:00–2:30pm, and by appointment

Course Assistant Gwyn Moreland ([email protected])Office Hours: Th 10–11am, F 3–4pm in EH 2nd floor commons

Class MTWTh 11:10am–12:00pm, 3314 Mason Hall

Website The course website can be accessed through CTools.

Course DescriptionOverview. Math 176 develops fundamental results in real analysis via the study of sequences and theirconvergence. We will especially rely on three central principles: the nested intervals axiom, iteration, andthe bisection method. These tools will allow us to define the real numbers, create sequences and series andstudy their properties, and investigate the fundamental concepts of topology (openness, closedness, andcompactness). We will then use these ideas to study continuous functions, derivatives, and integrals. Majortheorems that we will prove in this course include the Brouwer Fixed Point Theorem (concerning fixed pointsof a continuous transformation of the disk), Taylor’s Theorem (about approximating differentiable functionsby polynomials), and the Fundamental Theorem of Calculus (relating the processes of differentiation andintegration).

Format. This is an Inquiry–Based Learning (IBL) course. In order to support development of problem-solving skills, communication, and mathematical habits of mind, we will spend the majority of class timeworking in groups and presenting ideas and solutions to problems. I will give occasional mini–lectures to setthe context for the class activities, but my main role will be to support and facilitate your engagement withand exploration of the material. This means that we are jointly responsible for how class time is spent aswell as the successful development of the course!

Course ObjectivesWe will be developing some of the central results in real analysis, participating in the process of doingmathematics, and studying the disciplinary norms of mathematical writing and proof. More specifically, theobjectives of this course are that you will be able to:

• Write clear, succinct, precise, and logical mathematical solutions and proofs.• Determine the important ideas and processes in a solution or proof.• Evaluate your own understanding of a topic and ask questions to further your understanding.• Use mathematical language and symbols as a way to clarify and make intuition precise.• Make mathematical conjectures from data and examples.• Typeset mathematical exposition using LATEX.• Explain and prove some of the fundamental ideas and results of real analysis.• Describe the use of nested intervals, iteration, and the bisection method as tools in in topology andanalysis.

• Create a narrative through the basic components of real analysis, and relate and compare conceptsof sequences, topology, continuity, derivatives, and integrals.

Course ResourcesCTools. The CTools site is where announcements, worksheets, homeworks, and grade information will beposted. Make sure to reference it regularly.

Piazza. This term we will be using Piazza for online class discussion. The system is set up to efficientlyprovide access to classmates, the course assistant, and the instructor. Rather than emailing your questions,please post them on Piazza, where other students may benefit from the question and answer. Piazza alsoprovides a platform for peer collaboration and class–related discussion. Our class page is embedded in theCTools course site, and can also be found at: https://piazza.com/umich/winter2014/math176w14/home

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* Inquiry-Based Learning refers to a broad collection of student-centered active learning classroom techniques. This particular course traditionally uses in-class group work and student presentations. I have sought out train-ing in IBL classroom techniques from the Academy of Inquiry-Based Learn-ing, as well as by attending an MAA PREP Workshop.

Course Snapshot:Explorations in

Topology and Analysis

annotations syllabus student reflections

Math 176 is designed for students in the honors college who have taken AP Calculus, but who are not typically math majors (although I recruited some through this course!). I have taught this course for enrollments of 8 and 17 students. I approach this course as an opportunity to pitch mathematics and mathematical culture as something students will find both intellectually interesting and exciting.

- 3 -

** My syllabus sets a context for the course and makes my expectations of students explicit. The Course Objec-tives are an especially important part of this, and I ask students to re-visit these objectives periodically during the semester. These objectives also create a way to gauge whether stu-dents have learned what I want them to in the course, and they form a basis for how I structure class time, home-work assignments, and assessments.

The Course Objectives give students a way to gauge their own success in the course as well as a way for me to evaluate whether the course is meeting the needs of the students involved. I ask students during and at the end of the semester to reflect on whether they are making progress on the course objectives. This gives me feedback on what parts of the course are most central to it, what areas students need more support in, and whether the course objec-tives are clear. The following responses are taken from an end-of-the-term reflection asking students if they achieved the course objectives and, if so, how.

*

**

2

Sharelatex. Sharelatex is an online LATEX editor that we will use for collaboratively editing course notes aswell as typing up homework. One especially useful feature is that it allows multiple people to edit the samefile (even simultaneously!). We will spend some time at the beginning of the semester learning the necessarysyntax and exploring what LATEX can do.

Outside Resources. The IBL nature of this class is intended to provide a space for you and your classmatesto discover and create mathematics. For this reason, outside resources such as the internet, other textbooks,or people not associated with this class, are forbidden without explicit permission from the instructor. Thiswill be regarded as an issue of academic integrity.

Office Hours. I encourage you to attend office hours! See the office hours schedule above. Office hours area great place to spend extra time on the course material and help you build your understanding and skill,as well as get help with assigned work.

Assessment PlanYour course grade will be determined from the following categories and weights:

Participation 10%Textbook 20%Homework 25%Exams (2) 30%Final 15%

You are guaranteed an A (possibly ±) for a score of 90% or higher, a B(±) for a score of 80% or higher, andso on. These thresholds may be lowered (but not raised) at the end of the course, in order to align with thehistorical grade distribution for this course.

Participation. Participation is a central component of this course, and all students are expected to play anactive role in the class. Students will be assessed on the effort put into group work, volunteering regularlyto present solutions, and participation in discussion of other solutions.

• Attendance. Coming to class is extremely important. Not only does skipping class affect the student’sclass participation grade, it hurts the group with which the student is working. Thus, a student isallowed at most three unexcused absences from class. The instructor will decide what is an acceptableabsence on a case–by–case basis, and all absences due to illness require a note from University HealthService. For every unexcused absence beyond the third, two percentage points will be lost from theparticipation grade.

• Presentations. All students are expected to regularly present their (or their group’s) work and ideasin class. The goal of presentations is to further our understanding of mathematics via the exchangeof ideas—in particular, a successful presentation is not equivalent to the mathematics being correct.Mathematical missteps and attempts that don’t quite work out are often necessary in discovering acorrect solution, and one student’s confusion usually leads everyone to a better understanding of thematerial.

• Asking Questions. Asking questions and suggesting improvements in this class is very important tothe learning process in this class, and we will work to develop an atmosphere in which these activitesare natural and valued. Students are expected to be engaged in their classmates’ presentations andgive appropriate mathematical, wording, notation, or organization suggestions (these solutions willbecome part of the textbook!).

Textbook. The texbook is a set of course notes in which students will be asked to provide proofs andcomplete investigations and problems. For each unit (see the Topics section below), students will be putinto groups (“course notes group”) tasked with completing the course notes. The notes will be posted onsharelatex.com, and groups will need to type up solutions and proofs into the appropriate places. They areformatted in LATEX, a powerful typesetting program that is the standard for most professional mathematicswriting, and for which we will spend some time learning the syntax.

annotations syllabus student presentations

The problem below is a student proposed proof of the statement “a subsequence of a convergent sequence converges,” along with class-generated annotations and edits. There are a lot of defini-tions that students have to parse and then use while selecting appropriate notation, so we talked about the structure of the argument, improved notation, and drew some pictures to make sense of the notation.



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While discussing the problem above, we talked about the structure of the argument and whether it could be simplified (the notes in yellow are sum-maries of student input). Students were then re-sponsible for cleaning up the solution and typing it into the course notes.

Students write their solutions to problems on the board and then we discuss and edit them as a class.

* Students are active in my classroom, working in groups and presenting their ideas to the class. They receive feedback on their work from their peers as well as myself, and they also have the opportunity to think critically about their peers’ ideas. I find having students think critically about their own or peer-generated work in class also greatly increases their ability to make meaningful progress on prob-lems outside of class. By the middle of the semester, I am often able to give a lot of my feedback just by asking stu-dents what they think I will say or ask.

** I create a classroom culture where confusion and mistakes are produc-tively discussed, rather than hidden from view. This is in the service of both creating a level of comfort for students who are sharing their ideas publicly and advocating self-reflective learning habits. I believe that iden-tifying the difficult parts of an idea often leads to deeper understanding of the idea’s meaning and utility. One student commented on this classroom culture by writing, “It was fun to be in a class where risks were encouraged and mistakes were valued because they lead to further understanding.”

**

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Work for completing the course notes will take place both in–class and out–of–class:

• Worksheets will be done in class in groups (not necessarily the same as your course notes group).Problems on the worksheets show up as Theorems, Propositions, Definitions, Exercises, etc. in thecourse notes. Secretaries will be delegated (one per course notes group) to take thorough notes onthe solutions prepared in class and type these into the course notes in a timely manner.

• Other Theorems, Propositions, Definitions, Exercises etc. in the course notes will be assigned asgroup homework (in the course notes groups), and not discussed in class.

Wherever you work on the course notes, there are two commented lines:

%Typed by:

%Edited by:

Please add your name in the apprpriate line when you work on the course notes. Students will be evaluatedon the final group product as well as the extent of their individual contributions to the group.

Homework. In addition to out–of–class group work on the course notes, there will be individual homeworkassigned. Unless otherwise noted, you are allowed to work with other students in this class, but you may notuse outside resources (the internet, people outside of this class, textbooks, etc.)—this is considered an issueof academic integrity. For homework assignments, each student must submit their own individually writtenwork.

Exams. There will be two exams during the semester (approximately the middle of February and the endof March; you will have at least two weeks warning) with large take–home components. The final exam isscheduled for Wednesday, April 30 from 4–6pm. For take–home components of these exams, you will beallowed to talk with the instructor as well as use your course notes and other work for this class, but noother resources will be allowed (including communication with classmates).

TopicsThe following list of topics will be selected from based on student interests, instructor discretion, and timeconstraints:

Unit Sections

Sequences Real numbers and the Axiom of Completeness

Sequences and their properties

Series

Subsequences and the Bolzano-Weierstrass Theorem

Sequences in the plane

Topology and Open, closed, and compact sets

Continuity Continuous functions

Intermediate Value Theorem

Brouwer Fixed Point Theorem

Derivatives Defining the derivative

Mean Value Theorem

Chain rule

Higher derivatives

Power series and Taylor’s theorem

Integrals Defining the integral

Fundamental Theorem of Calculus

Anti-derivatives and evaluating integrals

Notes for Explorations in Topology and Analysis (Version Winter 2014) Instructor: D. Visscher

*Theorem 1.39. If the sequence (ak

) converges, then it is bounded.authors:

Rico, Theo, CarleeProof. Suppose (ak

) converges to a it L.

a1

a4

a5

a2

a3L

(

L− "

)

L+ "

The distance between ak

and L is bounded as k increases after a certain point. We can show thisby fixing " = 1 and applying the definition of convergence. In a

k

, 9N 2 N s.t. 8n > N, |an

−L| < 1.This splits a

k

into two sections a finite (n N) section and an eventually (n > N) section.

a1

a2

a3

. . . aN

finite

aN+1

aN+2

. . .

eventually

We then denote the finite section as the set F and the eventually section as a set E. E is aninfinite countable set that is bounded between L− " and L+ ". F is bounded because it is a finiteset. Therefore, the whole sequence is bounded by m and M , where

m = min{a1

, a2

, . . . aN

, L− 1}

M = max{a1

, a2

, . . . aN

, L+ 1}

So, therefore, if a sequence converges, then it is bounded.

authors:

Rico

*Remark 1.40. The converse of Theorem 1.39: If the sequence (ak

) bounded, then it converges.This statement is false, because some sequences are bounded, but never converge to a single value(e.g. a

k

= (−1)k).

However, bounded and monotone together do imply convergence. This is a very important theorem,and we will often use it as a tool to show that a given sequence converges. (Note that this tool isoften easier to use than the definition of convergence!)

Monotone Convergence Theorem. If the sequence (ak

) is monotone and bounded, then(a

k

) converges.

Proof. Let’s suppose that (ak

) is monotone increasing and bounded above byM (the proof is similarif (a

k

) is monotone decreasing and bounded below by m).

Idea: We will construct a sequence of nested intervals whose lengths go to 0, and invokethe Axiom of Completeness to get a real number L. Then we will prove thatits

k!1ak

= L.

Let I1

= [a1

,M ], and let c1

= a

1

+M

2

. In order to construct I2

, we consider two cases:

Case 1: c1

is also an upper bound for (ak

). Then let I2

= [a1

, c1

].

Case 2: c1

is not an upper bound for (ak

); then there must exist a natural number K such thatc1

< aK

. Let I2

= [aK

,M ].

This work is an adaptation of notes written by Alejandro Uribe.

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annotations syllabus student work

** The materials on the right show the process of creating the course text. After discussion in class, students were responsible for typing their solu-tions into the text on Sharelatex and peer-editing their work. Eventually, I comment on a draft. I am interested in students’ development as writers of mathematics, but also that they have complete and correct solutions writ-ten in the text. As for Theorem 1.39, I sometimes find that students still have some mathematical misconceptions that we need to address directly. It is then the students’ responsibility to re-vise their work and produce a clean, clear, and correct final product.



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* The primary objective of my exams is to assess whether students have achieved the learning goals I have set out. However, I also like to write problems so that students learn something from completing the exam. This is especially true for take-home exams. In this course, due to the small class size, I was able to have students schedule a time to turn in their exam and talk with me about their work. One student remarked, “I really ap-preciate that [Dan] took the time to sit down and talk to us about our exams before we handed them in. I learned a lot through talking through my work and discussing the challenges and successes that I had.”

***

1/23/2014 University of Michigan Mail - Math 176

https://mail.google.com/mail/u/1/?ui=2&ik=90387a4cd7&view=pt&search=inbox&msg=143c0e997c71c2a5 1/1

Daniel Visscher <[email protected]>

Math 176

Lola Thompson <[email protected]> Thu, Jan 23, 2014 at 4:02 PMTo: Daniel Visscher <[email protected]>

Dear Dan,

Thank you again for letting me sit in on your class today. Here is what I wrote in my notes:

0) Write definitions on the board before class

1) Announcements (5 minutes): Students submit homework via ShareLaTeX (each group turns in 1 file). Can finddirections on how to combine files in the Course Notes.

2) Minilecture on the definition of the limit of a sequence (5 minutes). Prof circles notation in definition andexplains what it means. Very clear! Draws picture to motivate the concept geometrically. Ends by explainingwhat the students will do today and connects their activities back to the definition that he has introduced. There'sa signup sheet for students to claim the problems.

3) Group work (30 minutes). Students work in groups of 2 or 3. Prof walks around the room to help. Everyoneseems to be ontask. Conversation waxes and wanes. 1 or 2 students seem like they don't understand what thequestions are asking them to do. Prof seems like he's trying to spend time with each group evenly. Conversationusually picks up after chatting with him. One group seems to finish 5 minutes before the others. Possibleobserver bias (groups talk more after I move to another table).

4) Going over solutions (10 minutes). Students write solutions on board but sit in their groups while prof stands atboard and discusses what is written. Prof corrects notation with colored chalk (great idea).

Please let me know if you have any questions about these notes.

I also wanted to let you know that the pair of guys who were sitting by the door started chatting with me aboutIBL after they finished their work. One of them mentioned that he likes IBL because it forces him to understandthings more deeply. The other one chimed in with similar sentiments but then pointed out that things that take theclass a week to figure out could be presented in a 10minute lecture. Both students agreed that they would likemore lectures but that they wouldn't want the lectures to take up the bulk of the class period. (You gave a reallynice explanation of the limit definition at the beginning of class, so I can see why they might want to hear morelectures from you!) I know it's still early in the semester and one of the students said that this is his first IBLcourse, so his opinion might change as he gets more comfortable with this format.

Thank you for pointing out all of those helpful resources during lunch. I'm very seriously considering usingShareLaTeX now that I know about it!

Best,

Lola

[Quoted text hidden]

annotations IBLobservation

peerobservation

midterm student

feedback

** In this particular class, students were in two groups of four. The ob-server noted questions (Q) and answers (A) provided by each student during group work, as well as my interaction with each group and with the class. My interaction with groups centers on ask-ing questions, which reflects my role as a guide for students who are building their own knowledge, rather than a distributor of knowledge that students are suppsed to retain.

* The observation on the far right was conducted by an instructor who was interested in how an IBL classroom runs, and it emphasizes how I orga-nize class and how I spend class time.



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Both observations on this page show that I devote a lot of class time to group work and discussion of solu-tions. As one student notes in the peer observation, it takes more time to cover material in this format than if I were to lecture. I think it is interesting that the same student did not want the bulk of class time to be devoted to lecture, but also mentioned that the IBL format forces him to understand things more deeply. I have found both of these comments are accurate for the IBL classes I have taught, although the pace usually picks up a month into the term as students become more comfortable with the format.

*

**

Course Information Math 385: Mathematics for Elementary School Teachers

Instructor Daniel Visscher ([email protected])3827 East Hall, Office Hours:

GSI Patricia Klein ([email protected])2848 East Hall, Office Hours:

Course Assistant Christine Shay ([email protected])Time and location by appointment

Class Section 1: MW 1:10–2:30pm in 4153 USBSection 2: MW 2:40–4:00pm in 4153 USB

Website The course website can be accessed through CTools.

Overview. How does one provide elementary–school appropriate justification of mathematical claims? Whydo the standard algorithms you know for addition, subtraction, multiplication, and division give the correctanswer? Can one intuitively make sense of the equation 5 − (−4) = 5 + 4? Can different representationsilluminate different mathematical concepts? Why is dividing by a fraction give the same result as multiplyingby its reciprocal? These are examples of the kind of questions we will consider in this class.

Our focus will be on understanding the mathematical structures, representations, and reasoning underly-ing numbers (whole numbers, integers, fractions) and their arithmetic (+,−,×,÷: strategies, models, andalgorithms). The goal is to create the specific mathematical knowledge needed for teaching: the knowledgenecessary to create instructive tasks for elementary school students and to understand and effectively utilizestudent thinking in the classroom. While this is a content class, and not a methods class, we will make thegreatest effort to explicitly connect what we do in our classroom to your own future classroom practice.

Format. This is an Inquiry–Based Learning (IBL) course. In order to support development of problem-solving skills, communication, and mathematical habits of mind, we will spend the majority of class timeworking in groups and presenting ideas and solutions to problems. I may give mini–lectures to set thecontext for the class activities, but my main role will be to support and facilitate your engagement with andexploration of the material. This means that we are jointly responsible for how class time is spent as well asthe successful development of the course!

We will be revisiting the mathematics of the elementary school curriculum as future teachers. In this capacity,we will work to develop mathematical practices that will be relevant in the classroom. More specifically, theobjectives of this course are that you will be able to:

• Communicate mathematical ideas effectively in both written and oral formats.• Find and evaluate mathematical content in elementary school work.• Persist in solving challenging problems and devise strategies to help in this process.• Justify mathematical solutions while considering both rigor and developmental context.• Use representations, models, and diagrams to help you solve problems and communicate solutions.• Recognize mistakes as an important part of doing and learning mathematics.• Explain your reasoning for using certain procedures in solving a problem.• Understand the ways in which math content knowledge supports the activity of teaching.• Evaluate your own mathematics as well as the work of your peers critically and supportively.• Learn from and mathematically evaluate material found in common teacher resources.

1

annotations syllabusexceprt

syllabus excerpt

Math 385 is the first in a two-semester sequence of math content courses for pre-service elementary school teach-ers. It typically enrolls students starting their third year at the university and first year in the School of Education. Class size ranges from 18 to 24.

* Mathematical Knowledge for Teach-ing is an idea that features promi-nently in the research of Deborah Ball. It is the professional knowledge one needs to teach elementary school mathematics, and it is comprised of six sub-categories. In this class, we focus on the area of Specialized Con-tent Knowledge. This is the content knowledge and mathematical skills teachers use to understand a student’s thinking, select illustrative examples, and instill mathematical practices in their classrooms.

4

The following schedule is subject to revision as needed:

I. Practices and Structures

0. Problem Solving...or, the art of doing mathematics.

1. Evens and Odds...or, the importance of good definitions.

2. Place Value...or, how we write numbers and the ways it affects our thinking.

II. The Four Operations of Arithmetic

3. Addition and Subtraction Strategies...or, a great wealth of patterns in our number system.

4. Integers...or, why we need more numbers and how to think about them.

5. Addition and Subtraction Algorithms...or, the close relationship between standard procedures and place value.

6. Models of Multiplication and Division...or, thinking about what our arithmetic means.

7. Primes and Factoring...or, the multiplicative structure of the integers.

8. Algorithms for Multiplication and Division...or, linking procedures and conceptual understanding.

III. Fractions and Decimals

9. Fractions, Models, and Equivalence...or, why we need even more numbers and how to think about them.

10. Decimals and Their Fractions...or, relating fractions with the place value system.

11. Addition and Subtraction of Fractions...or, how arithmetic plays with fractions (part 1).

12. Multiplication of Fractions...or, how arithmetic plays with fractions (part 2).

13. Division of Fractions...or, how arithmetic plays with fractions (part 3).

Research Study:“Comparing oral and traditional assessments in math content courses

for pre-service elementary teachers”

This course is part of a research project that seeks to investigate how assessments gauge student un-derstanding of course material and how they provide feedback. I plan to use this research to help me betterteach future elementary teachers and to enhance how future teachers learn the subject.

In this class, you will take both written and oral assessments, as well as reflect on the way you learn anddo mathematics via surveys and responses. (These are activities that I would normally do while teaching,regardless of my research.) I plan to analyze your work and your responses in these activities for my researchproject.

In any presentation of research results, I will not include your name or any identifying details. If I planto quote extensively (i.e., more than one sentence or an image of your work), I will follow up with an emailto check that this is acceptable to you.

Please contact me if you have any questions or concerns about this research.

Course Snapshot:Mathematics for

Elementary School Teachers

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** Course Objectives are a very im-portant part of my syllabus, as they are what I and students should measure the success of the class against. I ask students to reflect on whether they are making progress towards achieving these objectives in the middle and at the end of the term (see page 9).

*

**

Worksheet 2: Place Value...or, how we write numbers and the ways it affects our thinking.

MATH 385 (2014 Edition)

—Bill Watterson, Calvin and Hobbes

Learning Objectives

At the end of this section, you should be able to:

• Explain the difference between “number” and “numeral.”• Explain the structure of the way we write numbers, and in particular how it uses place value and

grouping.• Count objects in another base by utilizing the number line and/or appropriate grouping, and compare

this process to base-ten.• Order numbers in another base (e.g., place base-five numbers on the open number line), and compare

this process to base-ten.

Finer Sunbeam Vibes

In this worksheet, we will be studying the structure and patterns of the way we write numbers. Thisstructure underlies the way we will approach the processes, algorithms, and numbers that we will be studyingfor the rest of the semester. The problem is, the structure of this system is so internalized for us that it ishard to appreciate without some disruption.1

So, we will be developing our number sense in another number system. This is not because your futureelementary school students will need to know this system. This is because you will be teaching your futurestudents about the way we all record numbers (“base-ten”) and we want to disorient ourselves with base-fivein order to study the structure of such a system. We have all internalized our base-ten system so thoroughlythat it is second nature to us and we tend to equate a number (a cardinality, or how many objects thereare in a set, e.g., twenty-three) with its numeral representation in base-ten (e.g., writing a 2 followed by a3: “23”). These numerals are a recording system for numbers that facilitate processes like counting, adding,subtracting, multiplying, dividing, etc. But this is not the only way to record numbers.

The focus of this lesson is to study the structure of a base system; it is not to convert back and forthbetween base-five and base-ten. This means that we will work in base-five without referencing base-ten ordoing any conversions at all.

The Rules. In base-five we will record numbers with only five digits, which we’ll call A, B, C, D, 0. Thefirst few counting numbers in base-five are:

A, B, C, D, A0, AA, AB, AC, AD, B0, BA, BB...

In base-five we can only “count to D” before running out of symbols. So, as soon as we get a bunch of fiveitems, we group them together and write A0 to denote A group of five objects and 0 singletons. When weget another group of five, we then have B (our second digit) groups of five and write B0. Thus, the numeralDD represents D groups of five and D singletons. If we add one more, we’ll have five groups of five. Since wehave no digit to represent five, we will denote this new grouping as A00, which means that we have A groupsof “five groups of five”. In other words, we have A groups of twenty-five, 0 groups of five, and 0 singletons.Recall that in base-ten we have the 1’s place, the 10’s place, the 100’s place, etc. In base-five we have the A’splace (“one’s place”), the A0’s place (“twenty-five’s” place), then A00’s place (“twenty-five’s” place), etc.

1In the spirit of this worksheet, the heading of this section has been scrambled into an anagram.

1

2

Check your understanding

Exercise 1. Fill in the blanks:

A, B, C, D, A0, AA, , , , , , , , , , , , , , , , , ,

Exercise 2. How would you express the number of dots below in base-five?

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

Exercise 3. What base-five numeral comes after ACD? BDB? C0D?

Exercise 4. What base-five numeral comes before AB0? C00? ABC?

Investigations

Investigation 1. In your groups, get a bag of gummi bears, a few paper plates, and some pens (you candraw on the plates if you wish). You have two tasks. First, figure out how many gummi bears you have inbase-five using only base-five numerals and structure (recall the purpose of this worksheet). What processdid you use to figure this out?

Second, join with another group. Without starting over, figure out how many gummi bears your twogroups have together. Was there a process you used to help you figure this out?

Investigation 2. Make a very precise measuring stick with the following base-five numerals. Can you useany shortcuts you found in the gummi-bear activity translate to creating this measuring stick?

BA BC CD D0 DD A0A AAA ABC AD0 B00 ADD BBB C00 C0 A000 A0000

Investigation 3. What is BC + AA? Compute it and express the answer without converting to base-ten.

Investigation 4. What is the distance between D and BC? Compute it and express the answer withoutconverting to base ten. Hint: What is the distance between D and A0? What is the distance between A0 andBC?

annotations worksheet page 2

worksheet page 1

I wrote a blog post on this class entitled “Promoting confusion... and its resolution” for Michigan Math In Action. It can be found at http://sites.lsa.umich.edu/michigan-math-in-ac-tion/2014/10/15/promoting-confusion-and-its-resolution/



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* I include learning objectives on each individual worksheet, so that students know what they are sup-posed to be getting out of working on the problems. I reference these often when we discuss the problems, as the primary purpose of the worksheet problems is not to know the solution to that particular problem but rather to develop skills and understanding. In order to align assessments with the skills and understanding we work on developing in class, I copy these learning objectives to review sheets for the exams, and I write problems that measure students’ achievement of these objectives.

*

Dan Visscher Math 385 IBL October 2014

1

Timeline

2:40-2:41 Professor Visscher passes out worksheets and makes announcements

2:41-2:47 Introduction to factoring and prime numbers

2:47-3:22 Students work in groups on worksheets

3:25-3:35 Students share and discuss solutions to problems 1-3

Observations

1. Introduction to Topic: Professor Visscher provided a short introduction to prime numbers and

factoring, the topic covered on the worksheet for the day. He began by asking students to list the

factors of 10. Students listed the factors, but when asked about the smallest number of factors they

questioned why there were only two. Professor Visscher assured them that they would go deeper into

this concept as they worked on the worksheet.

2. Worksheets/Group Work: Students worked on choosing a definition for what constitutes a prime

number. Students decided that 1 was not a prime number after creating the prime factorization for the

number 18 (a question on the worksheet). Students based their reasoning on the idea that if 1 was

prime, the prime factorization would go on “forever,” and therefore 1 was most likely not prime.

Students discussed which integers listed from 1 to 150 were prime, and although most groups

explicitly stated that they were using divisibility rules, other groups did not explicitly state that as

their method of finding prime numbers; they simply went through the process. While working

students found it harder to test if integers were prime the closer they were to 150.

3. Student-Teacher Interactions: Professor Visscher and the assistant, Patricia, visited each group

multiple times. Students looked comfortable asking questions and discussing their ideas with

Professor Visscher and Patricia. Students asked questions such as “why wouldn’t 1 be prime” and

“how can I tell if a number is prime if it’s really large”? Students were given guidance on how to

answer these questions in their solutions such as considering the definition for “prime” they chose in

question 1.

4. Presentation of Solutions: Professor Visscher used the class roster to call on students to share their

answers. It was assumed that this was to randomize the process and give everyone a chance to speak.

Students gave their answers verbally for the first two questions, but for the third question a student

from one group was selected to present their solution to number 3 on the document camera. Students

were allowed to ask questions after the solution was presented, and overall groups were in consensus

about the answer.



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Communicate mathematical ideas effectively in both written and oral formats.• “Yes! I have definitely made progress here. For example, at the beginning of the semester I

struggled with making my explanations clear, because I did not explain things using defini-tions. Now, I automatically think of definitions I could use when I start my explanation.”

• “Yes. My workout problems have become more clear and explicit, with more dependence on diagrams to represent the ideas I write.”

• “No. I am still having trouble presenting mathematical ideas in a clear way. One way I might make progress towards this objective is to have other people read my homework before I submit it, and see if they understand what I am trying to say.”

Find and evaluate mathematical content in elementary school work.• “Yes. I have started to look immediately for the methods the student used instead of just

looking to see if it’s right or wrong. For example, on this week’s homework, I immediately started trying to find the student’s process instead of just looking at their final answer.”

• “Yes. In some of the homework, we are asked to evaluate and describe what students are doing to get their answers. I feel like I’ve improved. I’m able to see a problem, and figure out what they have done after analyzing their work. However, it takes me a while to do so.”

Persist in solving challenging problems and devise strategies to help in this process.• “No, I need more practice with this. I struggle the most with taking what I know and apply-

ing it to new problems - I am much more use to repeating what I have done before.”• “Yes. It takes me a while to completely understand what each question is asking and how I

can come to clear, concise solution to the problem. I make sure to visualize and use a hands-on approach to each problem. I work through the problems with my peers in some cases. Also, I scale down problems to make them easier to decipher.”

Use representations, models, and diagrams to help you solve problems and communicate solu-tions.

• “Yes, I have begun to use models/representations more frequently. I have always been more comfortable with just creating an equation or solving something algebraically, so I will still need opportunities to solve or understand problems with using representations.”

• “Yes. When I justified algorithms for homework AA, I used a model to show how bundling works in order to explain my thoughts. Before, I would have just wrote it out, and probably done so in a confusing manner.”

• “No. I am not a learner who is able to think about a problem through using models but would rather explain my understand through words. I like seeing models as they do help my understanding but i have a hard time constructing my own.”

annotationsstudent

mid-term reflections

IBL observation

In the middle of the term, I ask that students report on their progress towards the course objectives listed on the syllabus. This both reminds students about the learning goals for the course and gives me feedback on whether the structure is appropriately supporting indi-vidual students. Because many of the students are not initially familiar with this format of a mathematics class and have not had to previously engage with the mathematics in the way I ask them to, I find it particularly valuable to gather focused feedback in the form of students reporting on their progress toward well-defined objectives. The responses below are to a prompt asking students whether they feel they are making progress towards the listed objec-tive (yes/no), and to provide specific evidence or else an idea for how to work on this goal.

I actively seek feedback on my teach-ing and assess whether the methods I use are working for the students I have in class. The observation here was conducted by a School of Education graduate student whose focus was the Inquiry Based Learning classroom, and it gives a picture of how I interact with students during class. The mid-term student reflections give an indication of how students are meeting course objectives, as well as which objectives particular students feel they are strug-gling with or could use more support for.

* During this interaction, one student suggested that the smallest number of factors a whole number could have is two, and another student brought up the number one as a counterexample. In fact, the first problem on the work-sheet asked students to evaluate two possible definitions of “prime num-ber” based on the repercussions of including one as a prime number. So, I deferred this conversation to group work. In groups, students figured out that if one was prime, then the funda-mental theorem of arithmetic would have to be restated. This student-lead process is important in supporting the learning objective of evaluating and creating good definitions. It also gives students a more robust understanding of whether one is a prime number.

*

*

University of Michigan Winter 2014 Final Office of the Registrar - Evaluations 8 students responded out of the total enrolled 8

ro.umich.edu/evals/Instructor with Comments Report2014-04-10 - 2014-04-24 Report ID: MSR04734

Instructor: Visscher,DanielMATH 176 001

Date Printed:5/1/2014 22:40:07 PM Page 1 of 2

Other Users of This Item*

Responses from your Students** University Wide School/College5

SA4A

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YourMedian

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1 Overall, this was an excellent course. 5 2 1 0 0 0 4.70 3.95 4.33 4.73 3.75 4.00 4.332 Overall, the instructor was an excellent teacher. 7 1 0 0 0 0 4.93 4.16 4.60 4.85 4.11 4.50 4.833 I learned a great deal from this course. 8 0 0 0 0 0 5.00 4.00 4.38 4.72 4.00 4.13 4.504 I had a strong desire to take this course. 5 0 3 0 0 0 4.70 3.67 4.17 4.63 3.25 3.65 4.00

121 I gained a good understanding of concepts/principles in this field. 5 3 0 0 0 0 4.70 3.98 4.21 4.55140 I deepened my interest in the subject matter of this course. 6 1 1 0 0 0 4.83 3.89 4.25 4.63199 The instructor explained material clearly and understandably. 7 1 0 0 0 0 4.93 4.13 4.50 4.80200 The instructor handled questions well. 7 1 0 0 0 0 4.93 4.13 4.50 4.81217 The instructor treated students with respect. 7 1 0 0 0 0 4.93 4.50 4.78 4.90219 The instructor was willing to meet and help students outside class. 7 1 0 0 0 0 4.93 4.38 4.71 4.88229 The instructor used class time well. 5 3 0 0 0 0 4.70 4.14 4.50 4.80230 The instructor seemed well prepared for each class. 7 1 0 0 0 0 4.93 4.33 4.69 4.86240 The amount of material covered in the course was reasonable. 4 3 1 0 0 0 4.50 4.00 4.30 4.61256 Working with other students helped me learn more effectively. 6 2 0 0 0 0 4.83 3.67 4.00 4.50340 The textbook made a valuable contribution to the course. 1 1 2 0 0 4 3.50 3.33 4.00 4.38509 The instructor was concerned that we learn. 6 1 0 0 0 0 4.92 4.21 4.60 4.83891 The workload for this course was (SA=LIGHT...SD=HEAVY). 1 0 6 1 0 0 3.00 2.50 3.00 3.40892 Students felt comfortable asking questions. 7 1 0 0 0 0 4.93 4.17 4.50 4.79893 Graded assignments reflected the material covered. 4 4 0 0 0 0 4.50 4.08 4.36 4.67894 The grades in this course were fairly determined. 3 3 2 0 0 0 4.17 4.00 4.30 4.64895 Students' difficulty with the material was recognized. 4 2 1 0 0 0 4.63 3.88 4.14 4.50896 My expected grade in this course is (SA=A, A=B, N=C, D=D, SD=E). 6 1 1 0 0 0 4.83 4.38 4.67 4.86897 The course requirements were clearly defined. 5 2 0 1 0 0 4.70 4.17 4.50 4.72898 The instructor presented material clearly in lectures/discussions. 7 1 0 0 0 0 4.93 4.22 4.63 4.83

Written Comments

900 Comment on the quality of instruction in this course.

Student 1 I LOVED THIS CLASS. Great instruction and IBL was awesome

Student 2 Having never taken an IBL class before Math 176, I am very interested in taking another one in the future. As a person who is usually very shy, I find speaking in large class settings intimidating. The opportunity to discuss problems and concepts in small groups was invaluable to me and allowed me gain a lot out of each class. I really liked the freedom that we were given to explore and experiment with the topics learned in mini lectures. The thing that I liked most about this class was that it was never discouraging. We never were overwhelmed or overly frustrated when solving problems because the process of finding a solution was valued more than the solution itself. It was fun to be in a class where risks were encouraged and mistakes were valued because they lead to further understanding.





Dr. Visscher was an excellent instructor. He was really good at guiding us through problems in a way that allowed us to discover the answers for ourselves. His mini lectures were clear and concise. I also really appreciate that he took the time to sit down and talk to us about our exams before we handed them in. I learned a lot through talking through my work and discussing the challenges and successes that I had.

One improvement that I would suggest for Math 176 would be to have a longer class period, if possible. Often 50 minutes was not enough time to fully digest the information that we received in mini lectures and we were often cut off half way through a problem and struggled to pick it back up the following day. Another improvement would be to start derivatives and integrals earlier, if possible.

Student 3 NA

Student 4 I really enjoyed this course and found myself realizing that we really had learned a lot. Dan was a really great teacher and I wish I could have him again!!

Student 5 Dan was a very good instructor.

Student 6 NA

Student 7 NA

Student 8 Great course, instruction was clear. The course notes seemed complicated, but manageable once we got into the swing of things.

* The quartiles are calculated from Winter 2014 data. The university-wide quartiles are based on all UM classes in which an item was used. The school/college quartiles in this report are based on lower divisionclasses with an enrollment of 1 to 15 students in Division of Natural Sciences in the College of LS&A.** SA - Strongly Agree, A - Agree, N - Neutral, D - Disagree, SD - Strongly Disagree, NA - Not Applicable.







SA4A

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Student 3 NA



Student 6 NA

Student 7 NA


* The quartiles are calculated from Winter 2014 data. The university-wide quartiles are based on all UM classes in which an item was used. The school/college quartiles in this report are based on lower divisionclasses with an enrollment of 1 to 15 students in Division of Natural Sciences in the College of LS&A.

** SA - Strongly Agree, A - Agree, N - Neutral, D - Disagree, SD - Strongly Disagree, NA - Not Applicable.







Student 3 NA



Student 6 NA

Student 7 NA










Student 3 NA



Student 6 NA

Student 7 NA





ro.umich.edu/evals/Instructor with Comments Report

2014-04-10 - 2014-04-24 Report ID: MSR04734Instructor: Visscher,DanielMATH 176 001




SA4A

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annotations end-of-term student evaluations

Course Evaluations

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This is the official university course evaluation for my Explorations in Topology and Analysis course (Winter 2014). The response rate for this evaluation was 100%. The school/college quartiles given for comparison are based on similar courses (lower-division classes in the natural sciences with an enrollment between 1 and 15 students).

* “No comment” submissions have been collapsed.



SA4A

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121 I gained a good understanding of concepts/principles in this field. 7 10 1 0 0 0 4.30 3.95 4.25 4.60199 The instructor explained material clearly and understandably. 14 4 1 0 0 0 4.82 4.06 4.52 4.81200 The instructor handled questions well. 14 5 0 0 0 0 4.82 4.10 4.57 4.83217 The instructor treated students with respect. 18 1 0 0 0 0 4.97 4.58 4.81 4.92219 The instructor was willing to meet and help students outside class. 13 6 0 0 0 0 4.77 4.50 4.75 4.89229 The instructor used class time well. 16 2 0 1 0 0 4.91 4.17 4.57 4.80230 The instructor seemed well prepared for each class. 17 2 0 0 0 0 4.94 4.38 4.71 4.88256 Working with other students helped me learn more effectively. 10 8 0 1 0 0 4.55 3.72 4.00 4.50509 The instructor was concerned that we learn. 17 2 0 0 0 0 4.94 4.14 4.57 4.81510 The instructor was confident and in control of the class. 16 3 0 0 0 0 4.91 3.83 4.22 4.67891 The workload for this course was (SA=LIGHT...SD=HEAVY). 1 3 7 5 3 0 2.71 2.50 3.00 3.44892 Students felt comfortable asking questions. 12 7 0 0 0 0 4.71 4.21 4.58 4.81893 Graded assignments reflected the material covered. 7 10 1 1 0 0 4.25 4.13 4.46 4.72894 The grades in this course were fairly determined. 7 5 4 3 0 0 4.00 4.00 4.33 4.69895 Students' difficulty with the material was recognized. 8 10 0 0 0 0 4.40 3.87 4.17 4.50896 My expected grade in this course is (SA=A, A=B, N=C, D=D, SD=E). 6 10 2 0 0 1 4.20 4.36 4.69 4.88897 The course requirements were clearly defined. 7 12 0 0 0 0 4.29 4.17 4.50 4.75898 The instructor presented material clearly in lectures/discussions. 16 3 0 0 0 0 4.91 4.25 4.67 4.86


annotations end-of-term student evaluations

Course Evaluations

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This is the official university course evaluation for my Calculus II class (Fall, 2015). The school/college quar-tiles given for comparison are based on similar courses (lower-division classes in the natural sciences with an enrollment between 16 and 74 students).

Fall 2015 Final 19 students responded out of the total enrolled 23

University of Michigan Office of the Registrar - Evaluations

ro.umich.edu/evals/

* The quartiles are calculated from Fall 2015 data. The university-wide quartiles are based on all UM classes in which an item was used. The school/college quartiles in this report are based on lower divisionclasses with an enrollment of 16 to 74 students in Division of Natural Sciences in the College of LS&A.


* Comments with no or minimal con-tent have been collapsed. This evalua-tion in full is available upon request.

Written Comments


Student 3 Dan is an amazing prof. He has an incredible background in mathematics and can effectively share his knowledge to his students. He makes math interactive and enjoyable to learn.

Student 5 Calc is hard in college! I had to quickly understand that calc at Umich does not test what you know but your problem solving skills. Dan was the best though!! He helped us develop our "tool box" and his personality made lectures fun. His encouragement of a growth mindset helped me improve my confidence, especially with the quizzes. His outlook on class, class time, and learning really made a difference for me. Thank you Dan for an awesome semester!!

Student 7 Instruction is great. Dan really cares that we understand the concepts and do well. He is great at explaining complex ideas very simply and makes class fun.

Student 8 Great layout where the instructor touched upon topics for 20-30 minutes and then allowed students to work on a problem set together with the instructor floating from group to group. When the instructor was asked a question, he would often not give the answer, but prod the student to figure out and understand how the answer was achieved, a very beneficial tactic.

Student 9 Calc 2 is an extremely hard course but I'm surprisingly not TOO confused so overall I'm satisfied with what I've learned. I like the group activity because honestly if we had the standard do-problems-from-the-book for homework I don't think I would be able to and it would stress me out so much.

Student 10 The instruction was excellent. Expectations were clear and the professor, Dan, was an excellent professor who explained things clearly so everyone could understand the concepts

Student 13 Dan was wonderful at explaining even the most confusing concepts.

Student 14 Dan is a great instructor who is always looking for ways to better his instruction in order to help his students learn the material effectively.

Math 116 is coordinated centrally, with common exams that largely determine students’ course grade. The course coordinator also sets the sched-ule and assigns common homework, but otherwise individual instructors are entirely in charge of how to run their own class.

Course Evaluations: Student Comments

annotations student comments on evaluations

This is a compilation of selected com-ments from mid- and end-term evalua-tions both as an instructor and a teaching assistant. Comments are sorted by their primary content and cited by year, role, and course number in parenthesis. Evalu-ations are available in full upon request.

On classroom atmosphere:“daniel was great. really unintimidating and great about answering questions. i think i learned more from him than during regular lectures actually.” (2009, TA 300)

“I enjoyed working with others in small groups because we were all pretty motivated and able to stay on-task... I really respect Dan. He seems to genuinely enjoy his field and comes to class prepared and ready to engage with students and help them learn in class and during office hours.” (2012, Instr. 385)

“Having never taken an IBL class before Math 176, I am very interested in taking an-other one in the future. As a person who is usually very shy, I find speaking in large class settings intimidating. The opportunity to discuss problems and concepts in small groups was invaluable to me and allowed me gain a lot out of each class.” (2014, Instr. 176)

“The thing that I liked most about this class was that it was never discouraging. We never were overwhelmed or overly frustrated when solving problems because the pro-cess of finding a solution was valued more than the solution itself. It was fun to be in a class where risks were encouraged and mistakes were valued because they lead to further understanding.” (2014, Instr. 176)

“I really enjoyed this course and found myself realizing that we really had learned a lot. Dan was a really great teacher and I wish I could have him again!!” (2014, Instr. 176)

“Before this course I hatedddd math. I can’t even explain how grateful I am for having had this experience, because even though it was difficult, I feel 100 times more confident to teach math to my future students. This class has motivated me to teach math in a dif-ferent way than the norm.” (2014, Instr. 385)

“Honestly this has been one of the most productive and constructive math environments I’ve ever had the pleasure to learn in.” (2015, Instr. 116)

“Hi Dan! I really like the way you structure calc class, and it has become a class I look forward to! I find learning about the way people learn interesting, especially the growth vs. fixed mindsets...” (2015, Instr. 116)

On assessment:“...Daniel was overall a very good TA. He was accessible and helpful, grading and returning homework in a timely and well-annotated manner making the homework good for more than just points...” (2010, TA 340)

“Prof Visscher is an excellent lecturer and makes the course mate-rial relevant and interesting. For example, we submitted a paper on our choice of a real life example of a derivative. I’ve never heard of a paper in a math class before, but it really forced us to think about what we are learning and not just try to memorize material.” (2010, Instr. 220)

“I loved the seminar atmosphere. The small class size enabled us to work as a group and helped my learning. The grading was hard-espe-cially on the homework though it ended up just pushing us more. My writing in this class improved extensively because of this. The group homework while it was a good idea could be frustrating as some people in the group skipped all the meetings and failed to be a team player. Overall I loved this class, learned loads from Dr. Visscher, and will miss this class when we are finished.” (2013, Instr. 176)

“Dr. Visscher was an excellent instructor. He was really good at guiding us through problems in a way that allowed us to discover the answers for ourselves. His mini lectures were clear and concise. I also really appreciate that he took the time to sit down and talk to us about our exams before we handed them in. I learned a lot through talking through my work and discussing the challenges and successes that I had.” (2014, Instr. 176)

“...I believe that I learned a lot during the oral exams. This format of an exam allowed me to talk through what I knew and use the instruc-tor’s feedback to continue to build off of my ideas and expand them. I found that these exams were a helpful level of challenge and allowed me to present my knowledge in a unique and helpful format.” (2014, Instr. 385)

On classroom instruction:“...Math is actually interesting and patternful with Daniel.” (2010, Instr. 220)

“Visscher was really enthusiastic and the way he structured the class was incredibly effective for helping me learn the material. Usually I don’t spend much time on math classes because I don’t like it that much, but I actually had fun doing the work for this class.” (2011, Instr. 220)

“[Dan] was very good at communicating the concepts and principles behind the math so you get an idea of what the numbers represent. Person-ally I found this to be very helpful in learning Calculus on a more intuitive level rather than simply working through the mechanics of problems. He was also very approachable and always willing to help. He was one of the best math instructors I have ever had.” (2011, Instr. 220)

“The instructor showed an understanding of the content and did a nice job probing our thinking without giving us the answer.” (2012, Instr. 385)

“During class and office hours, Dan was always engaged in teaching. I appreciated that he and his assistant Will would circulate and stop by each group... I think this circulation was really advantageous as we worked on worksheets in small groups.” (2012, Instr. 385)

“I really liked the freedom that we were given to explore and experiment with the topics learned in mini lectures.” (2014, Instr. 176)

“Dan is very good at challenging his students to arise at their own conclusions in order to further their learning rather than just spoon feeding the answers. His allowance of healthy debate over topics was also helpful in strengthening my knowledge on topics.” (2015, Instr. 116)

“Dan was a great teacher. He helped us understand the most important parts of each section and we broke it up into doable chunks. Eventually, we were able to solve difficult problems systematically and easily with the practice gained. Questions were answered in a form that allowed the stu-dent discover the answer on his/her own, much like the Socratic method. I particularly enjoyed this because I felt like I was discovering patterns and that helped it stick more rather than reading it from the book.” (2015, Instr. 116)

On preparedness and organization:“[Strengths of instructor:] organization, desire to become a better teacher, strong technology/communication skills.” (2010, Instr. 220)

“Dan is an excellent TA, definitely one of the best in the Math dept. Extremely well-prepared, and he seems like he actually cared about helping us out.” (2010, TA 354)

“The instructor is very enthusiastic about the content and presents it in a very organized manner.” (2011, Instr. 220)

“I appreciated that Dan would write the agenda on the board because it let me know what would be going on in class. I appreciated also his writing on the board a schedule of due dates to keep us organized.” (2012, Instr. 385)

“Dan is a great instructor who is always looking for ways to better his instruction in order to help his students learn the material effectively.” (2015, Instr. 116)

Outside the classroom:“I just noticed that Daniel does not really give you the answers to your questions right away. He makes you do the work in front of him, which took quite a lot of time during his office hours.” (2009, TA 300)

“Dan was a great TA that communicated the material clearly, responded to e-mails and was very accessible to meet with outside of class.” (2010, TA 354)

“Professor Visscher is extremely well organized, and obviously loves math and is incredibly enthusiastic and knowledgeable about the subject. Additionally, he is almost always available, in person, by phone, or by e-mail, for help or instruction outside of class.” (2010, Instr. 220)

“He seemed dedicated to our learning and to the inquiry approach. He refused to ever just supply us with answers and instead would help to guide our thinking through suggestions or questions. His office hours were particularly helpful. Dan is really approachable and easy to talk to one-on-one.” (2012, Instr. 385)

“Dan is very invested in his students. He expects students to have read the day’s material before class, which allows him to spend most of class time helping students individually. He is helpful in office hours, and challenges you to make connections without just giving you the answers.” (2015, Instr. 217)

Courses Represented116: Calculus II (Michigan)176: Explorations in Topology and Analysis (Michigan)217: Linear Algebra (Michigan)220: Differential Calculus of One-Vari-able Functions (Northwestern)300: Foundations of Higher Mathematics (Northwestern)340: Geometry (Northwestern)354: Chaotic Dynamical Systems (North-western)385: Mathematics for Elementary School Teachers (Michigan)

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teaching portfolio - university of michigandavissch/cv/tp 11-2016.pdf · this teaching portfolio...

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