George Ashline, professor of mathematics and statistics, served again this year as a faculty consultant at the 2019 Advanced Placement (AP) Calculus Readings, held June 11-17 in Kansas City. Representing various universities, colleges, and high schools from around the world, about 1,170 faculty consultants graded the free response portions of about 445,000 AP Calculus exams during the week-long reading. Participating also in the four-day preparatory pre-reading before the start of the reading, George served as a table leader, co-facilitating the assessment work of a group of 16 readers, and a question team member, helping to prepare and implement the scoring guidelines and samples for one of the Calculus AB free response questions.

*(posted June 2019)*

**George Ashline**, professor of mathematics, recently co-published with Ian Adelstein the article “Reframing the Pythagorean Theorem” in the January, 2019 issue of *The College Mathematics Journal.* In the paper, George and his co-author review some of the history and proofs of the theorem and then provide two original framing proofs. Here’s the summary from the end of the paper: “The Pythagorean Theorem is one of the most fundamental and familiar results in mathematics. There are scores of different proofs of the theorem, and in this article we provide a history and overview of the framing proof technique. In particular, we offer two new hands-on framing proofs that extend some earlier well-known derivations.” George also organized a campus event on March 1 titled “Mathematics Education Forum: Alumni Reflections and Advice.”

*(posted June 2019)*

**George Ashline**, professor of mathematics, recently gave a series of invited talks/workshops to several different Geometry classes and about 120 students at U-32 High School in East Montpelier, VT. During two day-long school visits on Thursday and Friday, December 13 and 14, he presented the workshop “Framing the Proof of the Pythagorean Theorem” and the historical context for the Theorem. Through proof activities in each workshop, George led hands-on investigations by students at that high school of relevant mathematical and historical concepts. This visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. He also recently gave invited workshops to several classes/groups of students at Browns River Middle School in Jericho, Vermont. He led five separate 8th grade groups (about 125 students in total throughout the 8th grade) through the hands-on investigation/activity “Correlation Properties and Applications.” He also led several students in the BRMS math club through an investigation of the “Numbers on the Cards” activity. Each student received their own Purple Knights pen and set of stickers to recognize their participation in these learning activities. These visits were through Expanding Horizons. In yet another presentation, George gave an invited workshop to several groups of students at Mississquoi Valley Union High School in Swanton, VT. He presented the workshop “Exponential Functions in Snowflakes, Carpets, and Paper Folding” to three groups of four combined Algebra II classes. Through several activities in the workshop, George led hands-on investigations by about 50 students at that high school of relevant exponential function concepts and their application and a concluding exploration of the “Chaos Game”. (Also through Expanding Horizons). During the week of July 16-20, 2018, George co-taught the course Number Theory for Teachers as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development master’s program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. Twenty-seven teachers from across the state were actively engaged in the week-long Number Theory course. George served again this year as a faculty consultant at the 2018 Advanced Placement (AP) Calculus Readings, held June 2-8 in Kansas City. Representing various universities, colleges, and high schools from around the world, over 1,000 faculty consultants graded the free response portions of over 450,000 AP Calculus exams during the week-long reading. Participating also in the four-day preparatory pre-reading before the start of the reading, George served as a table leader, co-facilitating the assessment work of a group of fourteen readers, and a question team member, helping to prepare and implement the scoring guidelines and samples for one of the Calculus AB free response questions.

*(posted January 2019)*

**George Ashline**, professor of mathematics, recently addressed winners of the Vermont State Math Coalition’s Talent Search at the annual Vermont State Math Coalition Awards dinner. Similar to last year when he gave the invited presentation “The Pythagorean Theorem: A Look at its Rich History and Engaging Proofs," this year he presented “What Can’t You Do with a Degree in Mathematics?” Each year, the Vermont State Math Coalition sponsors a Mathematics Talent Search Program, in which a set of challenging math problems are posted on the VSMC website four times during the year and students across the state have several weeks to solve the problems and submit their solutions to the program director.

*(posted June 2018)*

**George Ashline,** professor of mathematics, in November gave six invited talks/workshops to different classes/groups of students at U-32 high school in Montpelier, VT. On Tuesday, November 21 and Thursday, November 30, he presented the workshop “Framing the Proof of the Pythagorean Theorem” to six different Geometry groups/classes. Through proof activities in each workshop, George led hands-on investigations by students at that high school of relevant mathematical and historical concepts. This visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. Earlier in fall 2017 gave several invited talks/workshops to several classes/groups of students at Browns River Middle School in Jericho, Vermont. He led four separate 8th grade groups (about 100 students in total) through the hands-on investigation/activity “Correlation Properties and Applications.” He also led two separate 5th and 6th grade groups (about 40 students in total) through the exploration/activity “Number Pattern Challenges”. Each student enjoyed receiving their own Purple Knights pen and set of stickers to celebrate their participation in these learning activities. These visits were sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. During the week of July 17-21, 2017, George co-taught the course Number Theory for Teachers as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. About 25 teachers from across the state and beyond were actively engaged in the week-long Number Theory course.

*(posted December 2017)*

**George Ashline**, professor of mathematics, recently gave invited talks to students at Vergennes High School and South Burlington High School. He presented “Encountering the Great Problems from Antiquity” to one group of AP Calculus students in Vergennes and two AP Calculus classes in South Burlington. Through activities in these presentations, George led discussion and investigations by students at those high schools of some relevant concepts in the History of Mathematics. This visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. George also gave the Faculty Showcase invited talk “My Evolving Use of Classroom Technology: Integration of Smart Boards, Classroom Captures, and Plickers” at Saint Michael’s College on March 3, 2017; and, he presented invited talk “The Pythagorean Theorem: A Look at Its Rich History and Engaging Proofs” at the Vermont State Math Coalition’s Annual Talent Search Awards Dinner at the Sheraton in Burlington on May 4, 2017. The ten award winners from a range of high school grades were in attendance, along with their families and teachers and other members of the VSMC. Other activities for George in recent months since last report: On November 1 and December 16, he presented “Correlation Properties and Applications” to two different AP Statistics classes at Colchester High School. Through several activities in the workshop, George led investigations by students in both classes of the definition and properties of the correlation coefficient and lines of best fit for pairs of quantitative variables. This invited presentation and visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. On November 19, he gave the invited presentation “Geometry’s Golden Treasure: The Pythagorean Theorem with Some History, Proofs, and Extensions” at the fall Northeastern Section meetings of the Mathematical Association of America at Trinity College in Hartford, CT. In the talk, George considered the historical background of Pythagorean triples and the Pythagorean theorem and examined a few derivations of the theorem and some of its extensions, including an interesting result due to ancient Greek mathematician Pappus of Alexandria. On December 20, George presented “Bias and Margin of Error” to one Statistics class at South Burlington High School. Through activities in the workshop, George led discussion of survey design and implementation and investigation by students of ways in which bias can impact statistical analysis and the types of error that can arise in a study. This invited presentation and visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(posted June 2017)*

**George Ashline**, professor of mathematics, recently gave six invited talks/workshops to about 125 students in different classes/groups of students at U-32 high school in Montpelier, VT. He presented the workshop “Encountering the Great Problems from Antiquity” workshop to five different Geometry groups/classes and the workshop “Exponential Functions in Snowflakes, Carpets, and Paper Folding” to one Pre-Calculus class. Through several activities in each workshop, George led hands-on investigations by students at that high school of relevant concepts in Mathematics. This visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. During the week of July 18-22, 2016, George co-taught the course Number Theory for Teachers as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. More than twenty elementary, middle level, and high school teachers from across the state and beyond were actively engaged in the week-long Number Theory course.

*(posted November 2016)*

** George Ashline,** professor of mathematics, recently was selected by the 2016 Northeast Section of the Mathematical Association of America Teaching Award Selection Committee to receive the 2016 NES/MAA Award for Distinguished College or University Teaching. The committee was “very impressed with George’s excellence in teaching and inspiration of students both at Saint Michael’s College and in greater Vermont." He was nominated by his Saint Michael’s Mathematics Department colleagues Zsuzsanna Kadas and Jo Ellis-Monaghan. The award presentation was to be at the Spring Meeting at the University of New England in Biddeford, ME June 3-4, 2016. As this year’s winner George will be asked to give an invited talk at the 2016 Fall Meeting to be held at Trinity College in Hartford, CT November 18-19, 2016. The Section also aims to nominate George for the 2017 Haimo Award for Distinguished College or University Teaching of Mathematics.

*(posted June 2016) *

** George Ashline****,** professor of mathematics, recently gave several invited talks/workshops to a wide range of classes/groups of students at Hartford High School in Hartford, VT and Vermont Academy in Saxtons River, VT and Whitcomb High School in Bethel, VT. At Hartford, he presented the workshop “Exponential Functions in Snowflakes, Carpets, and Paper Folding” to fourteen different classes (and nearly 200 students) in four groups during the four Hartford high school day blocks. At Vermont Academy, he led six classes in three groups through the activity “Number Pattern Challenges” and another two classes through the activity “Correlation Properties and Applications” and another two classes through the activity “Bias and Margin of Error” during both morning and afternoon periods. (There were about 100 students in all of the groups.) At Whitcomb HS, he led five groups of students through the “Estimating the Circumference of the Earth – Following in the Shadow of Eratosthenes” activity, in which students gathered sun shadow measurements at high noon and compared their calculations with those at a synchronized partnering location in Massachusetts to estimate the distance around the earth using an approach similar to a Greek estimation done in antiquity. (There were about 75 students in all of the groups.) These visits were sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(Posted May 2016)*

During the 2015-16 academic year, **George Ashline**, professor of mathematics, co-taught the fall course Calculus II for Teachers: Integral Calculus and spring course Probability, Geometry, and Measurement as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development Master’s level program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. About 30 K-12 teachers from across the state were engaged in the Calculus II course taking place over three weekends in September, October, and November. About twenty K-12 teachers from across the state were engaged in the Probability, Geometry, Measurement course taking place over three weekends in January, March, and April.

*(Posted May 2016)*

**George Ashline,** professor of mathematics, recently gave five invited talks/workshops to different classes/groups of students at U-32 high school in Montpelier, VT. He presented the workshop “Framing the Proof of the Pythagorean Theorem and Investigating Some Interesting Pythagorean Triple Properties” to two Geometry classes, the workshop “Exponential Functions in Snowflakes, Carpets, and Paper Folding” to one IMP 3 class, the workshop “Number Pattern Challenges” to an Algebra lab class, and a portion of the “Encountering the Great Problems from Antiquity” workshop to a group of advanced Geometry students. Through several activities in each workshop, George led hands-on investigations by students at that high school of relevant concepts in Mathematics. This visit was sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(posted May 2016)*

**George Ashline**, professor of mathematics, gave the Thirteenth Annual Pi Mu Epsilon lecture on Friday, November 6. His topic was “Exploring the famous problems from antiquity: angle trisection, cube duplication, and squaring the circle.” He covered three classical problems that Ancient Greeks wrestled with, explored nineteenth century insights about them and considered historical contexts for these problems along with interesting approaches to solving them using additional techniques and tools, including origami.

*(posted January 2016)*

**George Ashline**, professor of mathematics, during the week of July 20-24, 2015, co-taught the course Number Theory for Teachers as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. Thirty-three elementary, middle level, and high school teachers from across the state were actively engaged in the week-long number theory course.

George also was again this year a faculty consultant at the 2015 Advanced Placement (AP) Calculus Readings, held May 30-June 8 in Kansas City. Representing various universities, colleges, and high schools from around the world, faculty consultants graded the free response portions of over 424,000 AP Calculus exams during the week-long reading. George served as a question team member and table leader at the reading, helping to coordinate the implementation of the rubric and overall evaluation for one of the exam’s free response questions and co-facilitating the assessment work of a group of sixteen readers.On June 22, George engaged a group of more than 30 Vermont high school students in the presentation “Framing the Proof of the Pythagorean Theorem and Investigating Pythagorean Triple Properties” as part of the Governor’s Institute of Vermont Mathematics Institute, a week-long summer program for outstanding high school students, held at the University of Vermont.

On May 18, he presented “Correlation Properties and Applications” to one Algebra II class and “Bias and Margin of Error” to an AP Calculus class at Brattleboro High School. Through several activities in the first workshop, George led investigations by students of the definition and properties of the correlation coefficient and lines of best fit for pairs of quantitative variables. In the second workshop, students explored ways in which bias can impact statistical analysis and the types of error that can arise in a study. These presentations and visits were sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(posted September 2015)*

**George Ashline,** professor and chair of mathematics, during Spring Break week gave an invited talk/workshop to four different classes/groups of students at Canaan High School in Canaan, VT. He presented the workshop “Bias and Margin of Error in Statistical Analysis” to two Algebra II combined class group, followed by a Geometry and Pre-Calculus combined class group, followed by a large Middle Grades math group, followed by an Algebra I group. Through different hands-on activities in the workshop, George led investigations by students at this high school of relevant concepts in Mathematics and Statistics. These visits were sponsored by the Vermont State Mathematics Coalition’s “Expanding Horizons Program,” through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(posted April 2015)*

**George Ashline**, chair and professor of mathematics, participated in the Fall 2014 Vermont Council of Mathematics Teachers (VCTM) annual conference held at Saint Michael’s on Oct.17. The theme of the conference was “Teaching Mathematics Today." In one of the conference sessions, George presented “Encountering the Great Problems from Antiquity: Hands-On Trisection, Duplication, and Quadrature” and led participants through a consideration of straightedge and compass constructions and several problems which require additional methods, such as origami, to solve.

(posted November 2014)

**George Ashline**, chair and professor of mathematics, during the week of July 21-25 co-taught the course *Number Theory for Teachers* as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem-solving skills, and leadership skills. Thirty elementary, middle-level, and high-school teachers from across the state were actively engaged in the week-long course. **On June 23 and 24**, George engaged a group of more than 30 Vermont high school students in the presentation "Estimating the Circumference of the Earth – Following in the Shadow of Eratosthenes" as part of the Governor's Institute of Vermont Mathematics Institute, a week-long summer program for outstanding high school students, held at the University of Vermont. **On June 9,** George presented "Correlation Properties and Applications" to one AP class and "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to two other math classes at Brattleboro High School. Through several activities in the first workshop, George led investigations by students of the definition and properties of the correlation coefficient and lines of best fit for pairs of quantitative variables. In the second workshop, students explored exponential functions and their applications and the "Chaos Game." These presentations and visits were sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. **During Spring Break week**, George gave three invited talks/workshops to different classes/groups of students at Canaan High School in Canaan, Vt. He presented the workshop "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to one Pre-Calculus/Calculus II combined class, followed by an Algebra I class, followed by an Eighth Grade math class. Later during Spring Break, George gave four invited talks/workshops to different classes/groups of students at Milton High School in Milton, Vermont. He presented "Bias and Margin of Error in Statistical Analysis" to two AP Statistics classes, followed by the workshop "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to three Pre-Calculus classes and another version of that workshop to an AP Calculus class. Through several hands-on activities in each workshop, George led investigations by students at those two high schools of relevant concepts in Mathematics and Statistics. These visits were sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont.

*(posted August 2014)*

**George Ashline,** professor and chair of mathematics, recently gave four invited talks/workshops to different classes/groups of students at U-32 High School in Montpelier, VT. He presented the workshop "Bias and Margin of Error in Statistical Analysis" to one AP Statistics class, the workshop "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to two Algebra II classes, and the workshop "Correlation Properties and Applications" to an Algebra I class. Through several activities in each workshop, George led hands-on investigations by students at that high school of relevant concepts in Statistics and Mathematics. This visit was sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. *(March 2014)*

**George Ashline**, professor of mathematics, recently gave three invited talks to different classes/groups of students at Danville High School. He presented the workshop "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to a Problem Solving/Pre-Calculus class, followed by a Compound Interest lesson to a Financial Literacy class, followed by the workshop "Correlation Properties and Applications" to a Statistics class combined with a group of advanced students. Through several activities in each workshop, George led investigations by students at that high school of relevant concepts, and he also discussed college math requirements and some of the features/benefits of attending a college like Saint Michael's. This visit was sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. *(November 2013)*

**George Ashline,** professor of mathematics, recently presented "Exponential Functions in Snowflakes, Carpets, and Paper Folding" to two separate classes at Brattleboro Union High School on May 23. Through several activities in each workshop, George led investigations by students at that high school of exponential functions and their applications, and the "Chaos Game." These presentations and visits were sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. He was then invited by a different high school teacher to return to the Brattleboro school on June 4 to give another version of the "Exponential Functions" workshop to one class, and another presentation on "Bias and Margin of Error" to two statistics classes. Also, during the week of July 15-19, George co-taught the course Number Theory for Teachers as part of the Vermont Mathematics Initiative (VMI). The VMI is a three-year, statewide, comprehensive professional development program in mathematics for Vermont teachers to prepare them to become highly effective mathematics instructors with strong mathematics content knowledge, problem solving skills, and leadership skills. Twenty-three elementary, middle level and high school teachers from across the state were actively engaged in the week-long course. *(posted September 2013)*

**George Ashline,** professor and chair of mathematics, was again this year a faculty consultant at the 2012 Advanced Placement (AP) Calculus Readings, held June 8-16 in Kansas City. Representing various universities, colleges, and high schools from around the world, faculty consultants graded the free response portions of over 360,000 AP Calculus exams during the week-long reading. George served as a table leader at the reading, helping to facilitate the assessment work of a group of fourteen readers.

**George Ashline,** professor and chair of mathematics, on May 15, 2011 presented "Correlation Properties and Applications" to several high school mathematics classes at Bellows Free Academy High School in Fairfax, VT. Through several activities, George led investigations by students of the definition and properties of the correlation coefficient and lines of best fit for pairs of quantitative variables. This presentation was sponsored by the Vermont State Mathematics Coalition's "Expanding Horizons Program," through which college and university faculty are invited to give presentations and visit classrooms across the state of Vermont. Also, with several co-authors, George recently had an article accepted for publication as one chapter of the current NCTM yearbook. Here are the details of the publication: "Closing the Achievement Gap: Systemic Collaboration for Equity in Mathematics," (with Marny Frantz, Kendra Gorton, Sandy Hepp, and Stephanie Ratmeyer), In J. Bay-Williams and W. Speer (Eds.), *Professional Collaborations in Mathematics Teaching and Learning: Seeking Success for All*, NCTM 74th Yearbook, NCTM: Reston, VA (2012) 31-45. Also, George and Dave Landers, Instructor of Psychology, presented in an "Approaches to Learning with Tegrity" showcase November 29, 2011 at the campus Technology Center. Also, their presentations along with their students were recorded (via Tegrity, naturally) with the intent of making the recording available to the community via a Web site link thanks to the work of Jim Millard, senior instructional technologist. Over lunch, Dave and George and some of their students held a discussion of their experiences using Tegrity in a classroom setting, and how this technology tool has enriched classroom pedagogy and supported student learning. Dave and his students talked about peer and self-review in Sports Psychology, as well as how Tegrity is used for observation in Theory of Psychology. George uses Tegrity, clickers, and the Smartboard to help his students understand, apply, and review mathematical concepts in such classes as Calculus and Linear Algebra. Examples of successful Tegrity captures and some features of this technology were shared.