## A Community of Mathematicians: Using a Wiki in a Large Calculus Class

PCMI 2019 Workshop on Equity and Mathematics Education

2019 Organizer: Rochelle Gutiérrez, University of Illinois College of Education

Participants will further develop their understanding of equity (identity & power issues) in mathematics and consider how to expand our goals to rehumanize mathematical experiences for those with whom we engage. In this workshop, we will explore different perspectives/theories, reflect on our own practices, learn from experts in the field who have been altering their practices, and create our own action plans for work we intend to carry out after the workshop ends.

Ideal participants will include mathematicians, mathematics teachers, and mathematics education professors who have a specific project upon which they would like to focus. For example, you may have in mind a course you would like to alter in some way; a new initiative to launch; a summer camp or bridge program; a professional development or teaching activity to update; or simply a new way to think of assessments or evaluations. By the end of the session, you will leave with a more developed action plan and feedback from others so you can put your best foot forward in your future work.

## Denlow Public School

I visited Denlow Public School and did two workshops for the Grade 4 and 5 students. The Grade 4s played with probability, learned to play Pig. This simple dice game has been subject to a lot of deep analysis. Some folks at Gettysburg College have given an optimal solution to the game.

The Grade 5 students learned about Cat’s Cradle. They were very excited, and wanted to learn more. Many students already knew a figure or two. We covered Half Second Star, Cup and Saucer, and Jacob’s Ladder. I’m told that they’re still playing with the string that I gave them.

## NIBL&T Day 3

Team-Based Learning in a Large Calculus ClassHeather Bolles, Iowa State University; Amanda Baker, Iowa State University; Travis Peters, Saint John’s University; Elgin Johnston, Iowa State University; Darin Wohlgemuth, Iowa State University

Description: Implementing Team-Based Learning in a large-enrollment calculus class (more than 150 students) involves significant planning, stamina, and buy-in from students, instructors, and administrators. In this session, we share how we adapted the TBL flipped model to both Calculus I and II, the collaborative approach in developing and implementing materials, the continual evolution of the process, and the resources and classroom spaces we found helpful. Qualitative and quantitative data gathered over a three year period provide indications of success and identify where adjustments are yet needed. During the session, we will briefly model how we engage students in an application exercise following the Readiness Assurance Process, where students gain initial exposure to the topic.

Disseminating IBL via GeometryDavid M. Clark, SUNY New Paltz; Samrat Pathania, Wallkill High School

Description: This talk will report on a multiyear project to advance the use of IBL through the teaching of geometry. Currently the speaker is co-authoring a nearly completed book with Pathania entitled “High School Geometry: A Full Axiomatic Development”. Directed to college/university instructors, it will give them the full theoretical underpinnings of the speaker’s 2012 undergraduate text, “Euclidean Geometry: A Guided Inquiry Approach”.The 2012 text is primarily intended to give pre-service and in-service high school teachers (1) a personal learning experience through IBL and (2) an in depth understanding of exactly the topics they will need to teach. Whether or not they eventually do teach geometry, they will leave this course with a sound knowledge of what IBL is and how it is implemented so that they can draw on it for whatever they do teach.

High school students need to learn how to make evidence based arguments and judge when others are successfully doing so. This project will foster that goal by offering a vertically coherent view of geometry, giving high school students, high school teachers and research mathematicians a common basis for understanding this key subject.

Inquiry-Oriented Instruction as Principled ImprovisationDarryl Yong, Harvey Mudd College

Abstract: A very common IBL instructional routine involves posing tasks to students, monitoring their progress, and providing support as students tackle those tasks. The monitoring and supporting phases of instruction are often highly improvisational because they are less scripted and more dependent on what students do and say. What principles guide you during these phases of instruction? How can we work toward greater mathematical understanding, greater equity and inclusivity in our classes during this phase of instruction?

Oh! One more thing.

3D Printed Manipulatives for CalculusSebastian Bozlee, University of Colorado Boulder; Faan Tone Liu, University of Denver; Caroline Matson, University of Colorado Boulder; Cherry Ng, University of Colorado Boulder; Athena Sparks, University of Colorado Boulder; Porsche Adams Wootton, University of Colorado Boulder

Description: Intermediate students in calculus are often challenged by visualizing the applications of calculus in 3-dimensions. As we generalize curves in 2-dimensions and areas under a curve to surfaces and volumes constructed from functions, students benefit from a variety of approaches. We have used a 3D printer to develop numerous models and activities for our Calculus sequence. We will present an activity investigating solids with known cross-sections and various homework and lecture models for Calculus 2. We will discuss how these models are integrated in the classroom, how we developed the codes for and produced these models, and what plans are in progress to extend this work. We will also discuss how we established the 3D printer as a department resource including the expenses and support involved.

A brief description is available at: https://cu3d.github.io/

## NIBL&T Day 2

Oh man! Today was fantastic. I attended an online workshop by Tim Brzezinski on GeoGebra 3D. We did some 3D modelling, and did a worksheet. I’m amazed by all the stuff that GeoGebra can do. Previously, I’d only ever used it for simply graphing for my students. Now that I’m aware of its powerful geometric toolkit, it seems like the sky is the limit.

Towards the end of the session, we got to use GeoGebra 3D on our phones to do augmented reality stuff. Tim modeled a Toblerone bar, and then virtually super-imposed it on the real thing. Tim’s Geogebra page is full of amazing geometric stuff. He constantly tweets interesting math ed material at @dynamic_math. I encourage you to check it out. I’m definitely going to be using the augmented reality features soon.

Tim’s GeoGebra 3D with AR (Google): Explorations & Lesson Ideas

Steve Phelps’ Pythagoras Proofs without Words

The second session that I attended was a round table discussion on using inquiry to promote productive failure, resilience, grit, and growth mind set. We shared various metaphors that we use to explain growth mindset to our students. I often talk about math exercises as a form of weight lifting. Usually, I say this to make the point that watching me solve exercises doesn’t build up that skill in my students. The folks that I was chatting with developed the metaphor in a bunch of new directions, weight lifting involves: pain (productive failure), works best on a regular schedule, benefits from a minimum of expert guidance, prepares us for non-weight lifting tasks.

People have been giving five minute talks here. I love the format. You get to hear someone give a good pitch in a short, concise, and often simple way. The take away for me today was the notion of a MathsJam, a social gathering of people at a pub to talk about math. Brilliant!

## NIBL&T 2019 Day 1

I just finished up my first day at the Inquiry Based Learning and Teaching Conference in Denver, Colorado. It is great to see my friends from the Inquiry Based Learning Workshop last summer. My colleagues Alex Rennet and Jaimie Thind from UTM are here as well. A couple people at the conference have commented on what a strong IBL presence we have at UTM. Woot!

Setting the Stage for Small Group and Whole Class Discussions: Eliciting and Building on Student ThinkingKaren Keene, National Science Foundation & Nicholas Fortune, Western Kentucky University

Description: During this workshop, faculty will focus on two main themes: 1) how to set up their classrooms’ norms and environment to be able to have productive small group and whole class discussions, and 2) what teacher moves they can use during small group and whole class discussions to directly elicit their students’ thinking and build on that thinking. Each of these main themes will come with a mini-activity to gain first-hand experience. Topics for the first theme include but are not limited to ways to set up groups, how to provide an encouraging environment where it is acceptable to make mistakes, and using challenging tasks. Topics for the second theme include but are not limited to eliciting and building on students’ thinking, revoicing, peer to peer interactions, and connecting small group work to whole class discussion.

An active approach to calculus II and how it can help address (and create?) challengesJeanette Mokry, Dominican University; Aliza Steurer, Dominican University

Description: In addition to the new content that calculus II brings to our students, it also requires more decision-making and explanations of solutions than calculus I. Many topics in calculus II require students to make a decision. For example, “What series test should I use?” After deciding what series test to use, students must correctly interpret the results of the test and/or explain their reasoning. This can make the material quite challenging for students. Also, much of the content builds on prior knowledge, which can create challenges for the instructor, such as needing to present new material and also connect with the “old.” Mathematics also requires great attention to detail, including derivative notation, limit notation, and proper use of an equals sign. In the face of these challenges, how do we keep students motivated and help them see that, contrary to what they may have heard about the course, the material is doable? We will discuss how we have used active-learning worksheets to address some of these challenges as well as new challenges this approach can bring to the students and the instructor. At the beginning of our presentation, we will ask the audience what challenges they and their students have encountered with calculus II. Participants will also complete a short worksheet and, together as a group, we will discuss how that worksheet might address or create challenges in our classrooms.

From Place Values to Place Matters: An Indigenous Perspective on Calls for Diversity, Equity, and Justice in Mathematics and Mathematics EducationBelin Tsinnajinnie, Santa Fe Community College

Abstract: Despite perspectives that view mathematics as universal and culture free, policies and practices in mathematics education continue to perpetuate forces of settler colonialism and assimilation. Failed U.S. policies in Native American education illustrate the damaging impacts of assimilation and settler colonialism in education. What practices in our mathematics programs perpetuate settler colonialism and assimilation? In what ways can attenuating to our sense of place better serve goals of equity, justice, and inclusion?

## CMESG Day 4

Well, CMESG is wrapping up. It has been a good experience. Firstly, I am pleasantly surprised by the format of the study group. All weekend, I’ve been working with incredible people in math education to wrap my head around the notion of problem based learning. Everyday, we get to work together in small groups. The CMESG does not feel like a usual conference where one rushes from session to session.

Secondly, I am brimming with new ideas. There are lots of ew math education concepts to consider. I feel like I have a whole new vocabulary for thinking about what I want students to get from problems. My new project is to start workshopping some mathematically rich problems with students. My group has also asked me to write a comparison between IBL and PBL approaches. Are they mutually exclusive? Complementary? I’ll find out in Denver.

Today, I got to spend a lot of time with Rochelle Gutierrez, the second plenary speaker of the conference. She talked about her work “Living Mathematx: Towards a Vision for the Future” which attempts to theorize a more holistic ecological approach to mathematics. As for me, I really liked it and found that “living mathematx” resonated well with my thinking about teaching and learning. It tries to get at a kind of embodied ethics driven approach to mathematics which acknowledges other beings. I would say that my mentorship work is “mathematx” and my university teaching is “mathematics”.

## CMESG Day 3

Today at CMESG, we met with our working groups again. My working group on “problem based learning” started to design lesson plans around our problems. I was in the team of people working on upper-division problems. In particular, we wanted to design a lesson around the problem:

Classify the Platonic solids and prove that there are only five.

This turned out to be much larger than we expected, and the problem sort of blew-up in our face. We were not sure where to get started. It was a neat instance of what commonly happens when people approach a new problem; it gets out of hand and they’re not sure how to proceed. We struggled to figure out how much graph theory and group theory to introduce. Where would people take the problem?

If I were to run a problem based learning session, I’d like to go through a three levels of testing before trying the session with my students.

- Try the problem on a non-mathematical friend. Is this interesting?
- Experiment with some math friends. What content might it have?
- Take the problem to a math club as a lesson plan. Where do people take it?

Once I knew that the problem was intrinsically interesting, could have some mathematical content, and wouldn’t go too wonky, I would write it up as a lesson plan to be used in a real class.

## CMESG Day 2

We’re on Day 2 of the CMESG. I attended my first working group, on the theme of Problem Based Learning. This approach to teaching focuses on students’ experience of solving large open-ended tasks. Our working group is going to design a curriculum for “The Problem Based Learning University” which is a theoretical institution with 8000~10000 undergraduates, 500 graduate students, with “standard” service courses and no math program. We’re taking a problem based learning to problem based learning. I love this kind of meta-application of techniques.

We’d be teaching classes to:

- Engineering
- Commerce
- Life Science
- Humanities
- Pre-Health
- Arts
- Education

A good problem should be: “Real”, whatever that means.

Some other criteria that came up for us:

- Comprehensible (Language, culturally, student level)
- Investigative
- Interpretable
- Multiple paths & solutions
- Possibility of no solution
- Opportunity for meaningful failure
- Undirected and require independent thinking
- In class or long term term with research

For me, problem based learning requires bringing students an intrinsically interesting problem. I want to find problems that are engaging on their own. One good criteria for a worthwhile problem is that anyone who is curious would want to know how to solve it. Would my aunt want to solve this question?

How do sundials work? How could we build one?

The topic session that I went to today was: “*Culturally Sustaining Mathematics Education: Connecting Indigenous Knowledge and Western Mathematical Ways of Knowing*” given by Ruth Beatty (Lakehead University) and Colinda Clyne (Upper Grand District School Board).

## CMESG Day 1

Today the Canadian Math Education Study Group started up in Antigonish, NS. The purpose of CMESG is to get people in math education together in small working groups. Each group focuses on one specific “theme” in math education, and meets for three days to work on it.

The first event that I went to was an introduction to ‘for the learning of mathematics‘ a math education journal closely affiliated with the CMESG. It looks like a good venue for discussing ideas in math education, considered broadly. I am going to read through the issue that they gave me, and report back soon.

We also had a plenary presentation by Jean-Marie de Koninck, an amazing polymath from Laval University. He is a prolific researcher in analytic number theory, a world class swim coach, a math populariser, and more! He told us about the growth and development of Science and Math in Action. We got to hear how a change encounter with TV has grown in to a full time production of math shows for schools. The history about the gradual development of ShowMath 1 and 2 (for High Schools) and Le Petit ShowMath (for Elementary Schools) was fascinating. These are live performances involving a math professor and some professional comedians or clowns. It sounds amazing.

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