# Greater Upstate New York Inquiry-Based Learning Consortium

For several years a small group of professors in upstate New York have been meeting informally to share their enthusiasms, frustrations and triumphs related to the use of inquiry based learning in their mathematics classes. We are excited to announce that with generous funding from the Educational Advancement Foundation we are building this community in new and profound ways. Keep reading to find out what's going on and see how ** you ** can get involved!

## Where are we? What can we offer you?

We are all over the place! There are currently IBL practitioners at the following institutions in the greater upstate New York region:

Specifically, the consortium offers support and mentoring for those new to inquiry based learning and a supportive network for exchange of ideas for both novice and experienced users. For example, we offered both novice and more experienced users a workshop on Friday October 10, 2014, just before the Fall MAA Seaway Section Meeting, in Alfred, NY. We will continue building our network of IBL practitioners through our e-mail distribution list, the postings on this website, and dinner series events.

## What is going on in the consortium?

### IBL as independent study

One of the students in my spring 2014 IBL Introduction to Abstract Algebra class was keen to continue learning math using an IBL approach - so he undertook an independent study with me in Fall 2014. I will call the student Jim (not his real name). Given his interest in number theory and Abstract algebra, we chose as the basis for the study portions of the notes A Do-It-Yourself Introduction to Number Theory, by William Priestly, derived from notes by James Cross. (See JIBLM, No. 20, Dec. 2010.) These notes provide an introduction to number theory while simultaneously developing relevant ideas from Abstract Algebra. (Jim had worked with some, but not all of the Abstract Algebra ideas in his previous course- notably the independent study provided him the opportunity to explore rings.)

Doing an independent study based so firmly on IBL was a completely new experience for me. At first I had some concern about how slowly we seemed to be progressing through the material. However in retrospect, there is a huge difference between a student undertaking an IBL study on his own, and a class of students working on the material; Jim had to develop every idea on his own, without the benefit of other students to sometimes pick up the slack and guide the work forward. So, again in retrospect, I think the amount of material Jim ultimately mastered was amazing - and of course, this being IBL, by mastery I mean a deep and flexible understanding of the ideas.

We met for a minimum of one and a half hours each week. Jim came well-prepared with problems he had solved, and with questions on material that puzzled him. He maintained a careful notebook recording completed problems and key ideas.

The notes did a beautiful job of examining fundamental concepts of units, associates and primes, and how the existence of a Euclidean algorithm in a ring leads to unique factorization. The ideas were developed first in the very familiar context of the ring of integers, then generalized to the ring of Gaussian integers, and finally considered in polynomial rings. The notes also explored an extension ring of Z in which unique factorization does not hold. We ended the study by skipping to the last chapter in the notes to explore in more depth the characterization of primes in the Gaussian integers. At this point Jim independently and with no prodding from me developed an effective strategy for factoring a Gaussian integer into primes in the Gaussian integers; he derived a great deal of satisfaction from this - he got a real sense of being a mathematician.

As a final project Jim developed a 50 minute talk titled "Prime factorization in the integers and beyond" which he presented to his peers in the core mathematics class. He did an excellent job of selecting material accessible to his audience, and in crafting and delivering the talk. Jim obtained high praise for his obvious mastery and comfort level with the material.

In conclusion, a well-structured set of IBL notes and a willing and enthusiastic student are a winning mix for a highly successful independent study. Well done Jim!

Margaret Morrow, Associate Professor, Mathematics Department, SUNY Plattsburgh

### Visit causes contradiction?

Patrick Rault visited my Introduction of Proof class on Sep. 19, 2014. Students presented three tasks on proof by contradiction in a 50-minute class. One of my students was confused about which statement to negate in the beginning of a proof and had trouble deciding at the end what was actually proved. Patrick gave me the following useful advice on this. When one needs to show a theorem of the form P => Q by contradiction, instead of start the proof by saying

"For the sake of proof by contradiction, we assume that P is true and Q is not true",

one should start the proof by saying

"Suppose P is true. For the sake of proof by contradiction, let us assume that Q is not true."

The second version has the advantage that once a contradiction is reached at the end, students can immediately go to the sentence that has the word "contradiction" and be able to say that Q must be true (instead of trying to decide what to do with P).

After discussing my syllabus with Patrick, I also realized that I probably have over-worked my students by assigning too many homework. My students have daily homework, weekly homework, class presentations, portfolio, weekly journal and three exams. This makes me think about my objectives for each assignment and how I can re-design them to reach the same goal.

Xiao Xiao, Assistant Professor of Mathematics at Utica College

### Dinner Series: Buffalo

On Saturday, September 13th, three IBL practitioners gathered in Buffalo for a 3-hour lunch discussion. We discussed strengths in our courses and students, challenges which we've faced and ways to overcome them, and plans for the future.

One highlight was the idea that a lecture-based class usually has a "learning outcome for the class period," whereas in an IBL class we often think in terms of a "learning outcome for the time between class periods." Indeed, in an IBL class we might think of one unit as beginning halfway through a class period and ending halfway through the next class period. This is illustrated as follows.

Class period 1 | Class period 2 | |

Lecture: | Begin topic A, end topic A. | Begin topic B, end topic B. |

IBL: | End topic A, begin topic B. | End topic B, begin topic C. |

For this reason, it often makes more sense for a class observer to visit two classes in a row in order to see the entire progression of the topic. Though this is difficult for out of town visitors, local visitors should be encouraged to observe either two full classes in a row or two half classes in a row (i.e. the end of one class and the start of the next).

Please bring your own thoughts on this, or other IBL topics, to the upcoming MAA Seaway meeting. Or e-mail your own post for this BLOG to Ryan Gantner (for example, we would love to hear from someone about the Kenyon workshop or a mentoring trip). Or tweet using #UNYIBL.

The next planned dinner meeting will take place in Rochester on Tuesday 9/23 at 6:30pm. Contact rault@geneseo.edu if you would like to join us.

Patrick Rault, Associate Professor of Mathematics at SUNY Geneseo

### Calendar of Events

For questions or more information, please send e-mail to UNY.IBL@gmail.com .

The views and opinions expressed in this page are strictly those of the page author. The

of this page have not been reviewed or approved by Saint John Fisher College.