Swimming, Statistics, and the Normal Distribution

As we enter the final stretch of our accelerated summer statistics course, I find it hard to believe that there are only seven class meetings left, including the final exam. Summer courses move quickly, and each class brings us noticeably closer to the finish line.

Last weekend I had the opportunity to relax by the pool, recharge my batteries, enjoy some fresh fruit, and prepare for another busy week of teaching and grading exams. Those moments of rest are important during an intensive summer session, and they help me return to the classroom with renewed energy and enthusiasm.

This week I was pleased to see the progress my students are making. They recently completed their first exam and continue to gain confidence not only in their calculations but also in the language of statistics itself. It is always rewarding to hear students use statistical terminology naturally and to see them support one another as they work through challenging concepts.

Over the past few classes we have transitioned from studying binomial random variables to one of the most important probability distributions in statistics: the normal distribution. Students are now learning how to compute probabilities of the form P(X < x) when X follows a normal distribution, and how areas under the famous bell-shaped curve can be interpreted as probabilities.

The TI-84 calculator has become an important companion in this journey. Students are learning how to use its statistical functions to explore probability distributions and compute probabilities efficiently. Watching them become more comfortable with the calculator has been encouraging, especially as they begin connecting numerical results with graphical interpretations of data.

An interesting conversation this week with a colleague from Lehman College made me reflect on the role of technology in statistics education. She pointed out that in most real-world data science applications, datasets are often too large to enter manually into a calculator. Instead, data are typically imported from files and analyzed using software such as Python, R, or specialized statistical packages. In practice, a data scientist may simply load a dataset into a DataFrame, explore the data, create visualizations, and begin building models.

Yet I believe there is still considerable pedagogical value in tools such as the TI-84 calculator. While modern software is indispensable for working with large datasets, the calculator provides students with an accessible environment in which to explore probability distributions, random variables, and statistical concepts. By computing probabilities, visualizing distributions, and connecting numerical results to graphical interpretations, students develop intuition that will serve them well when they later encounter larger datasets and more advanced computational tools.

In a sense, the calculator acts as a bridge between the mathematical foundations of statistics and the computational tools used by practicing statisticians and data scientists.

Looking ahead, we will continue exploring the normal distribution before moving on to two cornerstone topics in statistics: the Central Limit Theorem and hypothesis testing. These ideas form the foundation of statistical inference and help explain how we can draw conclusions about populations using sample data.

The semester is moving quickly, but I am grateful for the energy, curiosity, and engagement my students bring to class each day. It has been a rewarding summer so far, and I am looking forward to finishing these last few classes together.

For now, it is back to teaching, grading, and preparing for another week of statistics. The bell curve is waiting for us.

Monitoring Arctic Change from Space: A New Student Research Opportunity

Today I had the opportunity to present a proposed undergraduate research project through the CREST undergraduate research program.

The project, Monitoring Arctic Change from Space: Remote Sensing Indicators of Permafrost Thaw Risk, introduces students to an interdisciplinary area that combines environmental science, remote sensing, data visualization, and data science. The project will explore publicly available environmental datasets related to Arctic change, including temperature, snow cover, vegetation indices, and other satellite-derived measurements.

Preparing this presentation was a rewarding learning experience. While my background is in mathematics, machine learning, and data science, developing this project allowed me to learn more about permafrost, Arctic environmental systems, and the growing role of satellite observations in monitoring environmental change.

Monitoring Arctic Change from Space: Remote Sensing Indicators of Permafrost Thaw Risk. Presentation prepared for the CREST Undergraduate Research Program. Image credit: NASA/GSFC/Jeff Schmaltz/MODIS Land Rapid Response Team.

I am excited to mentor a student through this project and to explore together how mathematical thinking, data analysis, visualization, and computational tools can help us better understand complex environmental processes.

One of the aspects I enjoy most about undergraduate research is that every project becomes an opportunity for both mentor and student to learn something new. This summer's journey will take us from mathematics and data science into the fascinating world of Arctic environmental data.

I look forward to seeing where this project leads over the coming months.

Statistics, Swimming, and the Road to the Bell Curve

The summer MAT 1272 Statistics course continues to move at an energetic pace. We are nearly finished with our study of binomial distributions and are about to transition into continuous random variables and one of the most famous probability distributions in all of statistics: the normal distribution.

Earlier in the course, we discussed the Empirical Rule as a way to understand how data are distributed around a mean. At the time, students had not yet learned about probability distributions. Now that students understand random variables and probability distributions, we are revisiting the Empirical Rule from a new perspective. Soon we will connect probabilities with areas under the Gaussian curve and begin using the normal distribution to answer meaningful probability questions.

Yesterday, students completed their first exam. I have also posted a comprehensive review worksheet that covers many of the topics we have studied so far, including basic probability concepts, conditional probability, simple linear regression, correlation, random variables, and binomial distributions. These topics provide a strong foundation for the material that lies ahead.

One of the most rewarding aspects of teaching this course has been watching students become increasingly comfortable with the language of statistics. They are beginning to use statistical terminology with confidence, ask thoughtful questions, and support one another during class discussions and group activities. It is encouraging to see students engage with ideas that initially seemed unfamiliar and gradually make them their own.

The class remains active and interactive. Students regularly stop the lesson to ask questions, seek clarification, or make connections between concepts. Those moments of curiosity and engagement are often the most enjoyable part of teaching.

Overall, this has been a wonderful start to the summer session. I feel grateful for such an enthusiastic group of students and for the opportunity to spend these weeks exploring statistics together. This weekend I plan to recharge with a good swim, catch up on a few projects, and enjoy plenty of fresh fruit before returning on Monday ready for another week of teaching.

It is hard to believe that only eight class meetings remain. There is still much to learn, but I am excited to see how much growth these students will continue to demonstrate in the weeks ahead.

From Counting to the Binomial Distribution

Today was another energetic day in my summer statistics course. We completed our introduction to the basic concepts of probability and began discussing probability distributions. Before reaching this point, we spent time distinguishing between discrete and continuous random variables, an important foundation for understanding how probability models are used in practice.

One of the highlights of the class was introducing the binomial distribution. After completing our discussion of counting techniques, including factorials, permutations, and combinations, I reminded two students of a simple experiment: tossing a coin repeatedly and counting the number of heads. It turned out to be the perfect example for motivating the binomial distribution. What had started as a counting problem naturally evolved into a probability model.

Throughout our discussions, I was encouraged by the students' comments and questions. From their participation, it became clear that many were beginning to understand the computation of a conditional probability and how is linked to determiening if two events are independent or dependent. These are subtle concepts that often require time and practice to develop, so it was rewarding to see students applying the ideas correctly in examples and classroom discussions.

The students also took a quiz on box-and-whisker plots. This assessment included identifying outliers using inner fences. I am not yet sure how many students successfully identified all the outliers, but the results will help me determine what concepts may need additional review.

There is rarely a dull moment in this class. Today we worked together with the TI-84 Plus calculator, computing factorials, permutations, and combinations. We then began calculating expected values and standard deviations for the binomial distribution. It was satisfying to see students connect the formulas with actual computations and interpret what the results mean in context.

As class was ending, I asked a couple of students whether they felt they were learning something from the course. They responded that they were learning a lot. It seemed like a genuine comment, and I certainly hope it was. More importantly, I can see evidence of their growth through their participation, engagement, and the thoughtful questions they ask during class. Students who were initially hesitant are becoming more comfortable contributing to discussions and working through problems in front of their peers.

Teaching an intensive summer course is always challenging because of the pace, but it is rewarding to watch students make connections across topics. In just a short period of time, we have moved from descriptive statistics and boxplots to probability, counting techniques, random variables, and probability distributions. I look forward to continuing this journey as we build toward more advanced statistical models and applications.

Teaching Statistics in an Accelerated Summer Course

Over the past two weeks in my MAT 1272 Statistics course, we have covered a wide range of foundational statistical ideas while balancing computation, interpretation, and technology. Because this is an accelerated summer course, I have been focusing on helping students build intuition through examples, visualizations, board work, and hands-on calculator practice.

We began with the fundamentals of statistics, including populations and samples, qualitative and quantitative variables, and the distinction between descriptive and inferential statistics. Students then learned how to organize and visualize data using frequency tables, bar graphs, histograms, stem-and-leaf displays, and box-and-whisker plots. Along the way, we discussed measures of center and spread, including the mean, median, mode, range, variance, standard deviation, quartiles, percentiles, and outliers.

More recently, we shifted our attention to relationships between two quantitative variables. Students learned how to construct and interpret scatterplots, determine whether a linear trend is present, and identify whether the trend is increasing or decreasing. We discussed how a regression line can be used to model data and make predictions, as well as how the slope and y-intercept describe the behavior of the model. Using the TI-84 Plus calculator, students learned how to compute the regression equation and the correlation coefficient r, which measures the strength of a linear relationship between variables.

Our next unit focused on probability. We studied basic probability concepts, conditional probability, independent and dependent events, the multiplication and addition rules, and counting techniques such as permutations and combinations. Throughout the unit, students used their TI-84 Plus calculators to perform calculations while developing a deeper understanding of how probability can be used to model uncertainty and real-world decision making.

The next chapter introduces one of the most important ideas in statistics: probability distributions. Students will learn about discrete random variables, probability distributions, expected value, and standard deviation for random variables. We will then study binomial and hypergeometric probability distributions and use technology to compute probabilities in realistic scenarios.

One aspect of this chapter that I am especially excited about is helping students visualize probability distributions. By constructing histograms of binomial distributions, students can begin to see how these distributions become increasingly bell-shaped as the number of trials grows. This observation provides a natural bridge to one of the central topics in statistics: the normal distribution.

Throughout the course, my goal has been to connect formulas, graphical representations, technology, and real-world applications so that students can see statistics as more than a collection of calculations. Whether analyzing data, modeling uncertainty, or making predictions, statistics provides a powerful framework for understanding the world around us.

Two Days into a Four-Week Statistics Course

June in New York City, on the way into an intensive summer teaching rhythm.

This summer I am teaching MAT 1272, an introductory statistics course compressed into just four weeks. Sixteen class meetings. One entire semester packed into sixteen mornings.

At the same time, I am trying to keep research projects moving forward before July conference deadlines arrive. Balancing teaching and research is always part of academic life, but summer courses have a unique intensity. There is very little time to waste, and every class meeting matters.

What is remarkable is that we are only two days into the course.

Just two class meetings.

Yet it already feels as though we have covered a tremendous amount of ground. In those first two days, we have discussed descriptive statistics, statistical inference, variables, populations and samples, parameters and statistics, qualitative and quantitative variables, and discrete versus continuous variables. We have also started organizing and visualizing data through bar charts and histograms. These may seem like simple topics, but they form the foundation for everything that follows in statistics.

Before students can analyze data, they must learn how to describe it, organize it, visualize it, and communicate what it means.

A quiet moment between teaching, research, and the pace of summer in the city.

One of the highlights of the course has been the group work. Students come from a wide variety of backgrounds and experiences, and it has been exciting to watch them engage with one another's ideas. Discussions often take unexpected and interesting turns. This week, for example, students have already begun grappling with the distinction between a sample statistic and a population parameter. They are starting to understand that while we may know the average of a sample, the true population mean often remains unknown.

Those are important moments because they signal the beginning of statistical thinking.

The first day of class also brought an encouraging surprise. A student who had previously taken the course approached me at the end of class to introduce herself and tell me that she appreciated my teaching methodology. As instructors, we never really know what students will remember from previous experiences, so moments like that are especially meaningful.

This summer I have relied heavily on the blackboard. There is something timeless about developing ideas step by step in real time, allowing students to see the reasoning process unfold. Eventually I will project visualizations and technology-based demonstrations, but for now the old-school approach seems to be working well. Chalk, dust, and all. The students seem to appreciate it, and the classroom conversations have been lively and engaging.

Chalk, dust, and statistics: the old-school classroom is still alive and well.

At some point, the TI-84 Plus calculators will make their appearance in the classroom. For now, however, the focus is on concepts, language, and interpretation. I want students to understand what a histogram represents before asking a calculator to create one, and to understand the difference between a sample statistic and a population parameter before pressing a button to compute a numerical value. Technology is an important tool, but it is most effective when it supports understanding rather than replacing it.

I've also been fighting a cold and have spent much of the week feeling under the weather. Teaching an intensive four-week course leaves little room for slowing down, however, and the positive energy in the classroom has made it easier to push through.

Perhaps my favorite part so far has been getting to know the students. I am still learning names, and they have been amused by some of the memory tricks and mnemonics I use to remember them. There is a good atmosphere in the classroom. Students seem appreciative, engaged, and willing to participate.

What makes me happiest is hearing them begin to use statistical terminology naturally. Words such as population, sample, parameter, statistic, qualitative variable, quantitative variable, histogram, and distribution are already finding their way into classroom conversations. These may seem like small victories, but they represent the beginning of statistical literacy. Students are asking thoughtful questions, debating ideas during group activities, and becoming more confident in their reasoning.

Tomorrow is already our first quiz.

That fact alone captures the pace of a four-week summer course. We have only met twice, yet students have already learned a substantial amount of new material and are preparing to demonstrate their understanding.

Time moves quickly for all of us. One student will be graduating next week, while another will be starting a master's program at the Graduate Center this fall. Other students are balancing work, family responsibilities, and their own academic goals. It is a reminder that every classroom brings together people at different stages of their journeys, each moving toward something new.

Before long, these sixteen class meetings will be behind us as well.

For now, I am grateful for a classroom full of curious students, thoughtful discussions, and the opportunity to spend June exploring statistics together. Only two days have passed, yet the classroom already feels alive with questions, new ideas, and aspirations for the future. It has been an exciting start to the summer.

A Classroom Built on Growth - MAT 1190 (Spring 2026)

As the semester comes to an end and tomorrow’s final exam approaches, I have been reflecting on Math 1190 CO. Independently of the grades my students will receive, this semester has been another reminder that I am not only a mathematician who teaches at the college level, but also a mathematician who continues learning through teaching each semester.

This class became a space where I witnessed growth in many forms. I saw a group of students who showed up regularly, participated actively, and genuinely engaged with the class activities. I watched communities form inside the classroom. I saw students helping one another, discussing ideas together, and becoming increasingly comfortable sharing their thinking openly.

Throughout the semester, we explored many topics: voting theory, probability, statistics, correlation, the normal distribution, z-scores, counting techniques, growth models, compound interest, and linear models. Beyond computations, we focused on interpretation, discussion, and applying mathematics to real situations. Some of the conversations that emerged from these topics were thoughtful, honest, and surprisingly rich.

Yesterday, during our final review session, there were still several review problems left to discuss. Instead of simply presenting solutions myself, I asked students to come to the board and present problems to the class so they could take a more active role in preparing for the final exam. What followed was one of those moments teachers remember.

One student went to the board for the first time all semester. I could see the satisfaction and confidence that came with simply trying. Two students worked together at the board as a team, discussing the exercises openly with the class while I stepped aside and observed. There was collaboration, openness, laughter, and genuine engagement. I saw students explaining ideas to one another instead of waiting passively for answers.

What stayed with me most this semester was not perfection, but growth. I saw integrity and accountability when students admitted they had not studied enough or had made mistakes. That level of honesty reflects maturity and a sincere desire to learn and improve.

One student told me she was actually enjoying studying mathematics. I told her that was a win for both of us.

I will miss this group.

To my students from this semester: thank you for showing up, for trying, for participating, and for contributing to the atmosphere we built together in the classroom. If you ever want to stop by my office just to say hello, please do. I would truly love to be present one day at your graduation. It was a privilege to teach you.

Now I will be shifting gears for the summer — returning more intensely to research, conferences, and catching up with that side of my profession, while also teaching an intensive summer course.

For the students who kept showing up even when things felt difficult: that matters. There is real power in continuing to show up, especially when something does not come easily.