A Model Mathematics Course: Eco-Math Link through Linear Algebra

Biodiversity and the Hudson River

Flowing from the Lake Tear of the Clouds, North the Hudson River journeys 315 miles and drops 4,322 feet in elevation before emptying itself into New York Harbor. The Hudson River is home to diverse populations of fish, birds, and mammals that cohabit and compete among themselves for resources. Recently the American shad, Atlantic sturgeon, river herring (blue back herring and alewife), American eel, and largemouth bass are in decline. Intense economic harvesting pressure and overexploitation cause coastal and marine species to decline. Therefore, harvesting and fishing should be managed properly and carefully to avoid decline of current population. Food web analysis provides important information regarding the nature of competition among various organisms.

Cluster analysisin graph theory is a popular method to seek partition of a given data set into several clusters so that the data points within the same cluster are more similar than those belonged in the separate clusters. In this project we will use cluster analysis using the concepts of linear algebra to study the competition among various species in a given food web, in particular, competition among various Hudson River species. Students will find a partition of the competition graphs based on the Hudson River food web such that the strength of competition (for shared preys) between two clusters (two groups of predators) is as low as possible; however, the strength of competition within the same clusters is as high as possible.

Big Ideas behind this project

Study and analyze Hudson River Food Web and its competition graph to interpret the strength of species competition. Upon completion of this project, students should be able to answer the following questions:

  • Which predator species are more connected than others?
  • What happens if a specific species (particularly, a prey) dies out? Particularly, how does the removal of a particular species affect its predators and also the overall competition among all predator species?

I believe through this project students will gain some insights to the mechanisms of interactions and competition among various species. Students will be able to propose further measures for early intervention if any species dies out, share their knowledge, and create public awareness of the need to promote a healthy and balanced ecosystem in their own community.

My goals as the course instructor are:

  • To assist students develop a deep understanding of core mathematical concepts and help them appreciate the usefulness of mathematics to analyze and explain their community and environment.
  • To create challenging environment for high achiever students.
  • To provide training in conducting research in an interdisciplinary field combining mathematics and ecology based on biodiversity of the Hudson River Estuary; a topic that is carefully chosen to hold students’ interests.
  • To motivate students in higher studies in an interdisciplinary field.
  • To help students retain knowledge for long term.

Students Learning Outcomes

  • To solve systems of linear equations using matrices.
  • To identify and use vector properties (spaces, subspaces, bases, inner product).
  • To identify properties of matrices (inevitability, eigenvalues, eigenvectors).
  • To use computer technology to solve practical problems.
  • To learn how to collect data.
  • To learn how to apply core mathematical concepts (particularly eigenvalues and eigenvectors) in solving real-world problems.
  • To understand interdisciplinary approach and the significance of it in real-world applications.
  • To write technical reports and disseminate the key findings.
  • To understand how to present research findings.
  • To learn how to work as a team.

How this will be done?

  • Teach core concepts from Linear Algebra; emphasize usefulness of eigenvalues and eigenvectors.
  • Require students to prove few theorems, lemmas and propositions regarding eigenvalues and eigenvectors.
  • Require students to review literature.
  • Introduce basic graph theory concepts and its applications to food web and species competitions.
  • Require students to visit website and show them how data is collected.
  • Require students to go to a Hudson River Fund seminar.
  • Require students to verify few theorems using the Hudson River data.
  • Introduce spectral clustering method.
  • Require students to analyze the Hudson River competition graph using spectral clustering technique.
  • Require students to interpret the results and make conclusions based on their findings.
  • Require students to write a technical report or present their findings.
  • Require students to produce fliers to be distributed later in high schools or local clubs in their own neighborhood.


I will assess the course and the project outcome as follows:

  • Individual student’s grading will be done based on a rubric. The rubric will be created based on learning objectives.
  • Report and final presentation will be graded.

    FoodWeb and the Competition Graph

  • Long term follow up will be carried out with interested students.
  • A pre-test and post-test survey will be administered.


An Idea on How to Incorporate OpenLab into your Course

I came across this article and thought that it might be beneficial to the Fellows. After last Friday’s powerful workshops I too am seeking new ways to include high impact practices into more of my teaching. This article focuses on “blended learning” and should work well for those of use who are trying to incorporate the OpenLab into our courses.

Let me know what you think.

“How Blended Learning Works” by Jeffery Galle

Your Research and Pedagogy Liaison — I’m here for you!

As the research & pedagogy liaison between the Brooklyn Waterfront Research Center and you, the Living Lab second-year fellows, I’m here to help you with making connections to community resources, designing research assignments, bringing data visualization and web mapping tools into the classroom, finding appropriate readings, and anything else related to your work developing templates for high-impact research assignments. I’ve reviewed the 4 groups’ templates for the seminars you are developing for one another and I have many ideas for supporting the research and activities you’ve outlined so far, so do not hesitate to get in touch with questions.

If you haven’t already, feel free to join the place-based learning toolkit on the OpenLab and add your own activities, resources, or assignments.

Looking forward to working with you all!

~Anne Leonard aleonard[at]citytech.cuny.edu


My Comments About Richard Feynman’s Book

The life of Richard Feynman has many interesting events. However, his life generally represents in my opinion, the direction many young male students with significant interest and experience in mathematical studies are invited to participate in. What I am speaking of is Physicists who work within the government; and who eventually end up working on top-secret projects in atomic-physics for national defense. Based on the information presented in the early part of his book, Mr. Feynman describes events, which lead me to believe in advance that he would eventually end up working with intelligence. The fact that he mentions working on specific government projects, which require a top secret security clearance was in my opinion, a clear indication of my previous expectations of the book.

The most important thing I came away with as a result of reading this book; is the journey one takes in obtaining knowledge. Mr. Feynman from as a young child consistently was in search of knowledge with a strong desire to learn more and solve complex problems. As the well-known musician and trumpet player Freddy Hubbert once said, “Embrace the journey.” Feynman as a young man began to embrace the journey of life, seeking knowledge of many things. I was also compelled to reflect on the Ontological Argument of Descartes, and the Fifth Meditation of Descartes, which I feel represents the development of knowledge through exploration of philosophical principles of learning. How we know what we know is a journey of trial and error of social development. The ethical issue Feynman speaks of at the end of his book is an idealistic view, which is generally non-existent or locked in the chains of our communications and involvement with others; particularly in employment. People most of the time will trade the truth, which lives in their hearts rather than run the risk of rejection.

Scavenger Hunt

Group Members: Melanie, Ralph, Muhammad

I had a lot of fun on the hunt, it has been a while since I walked the area.  I have worked in the area inspecting soil borings and foundations.  One of the things I love most about being an engineer is being able to walk around and point out projects that I’ve worked on, it gives me a sense of accomplishment.  I love maps and my sense of direction so this exercise was great for me! – Melanie

I did not know that we had show much to see within our neighborhood!!!! -Muhammad

After more than four years at City Tech, I finally got to see parts of Brooklyn Heights!!! Thank you Living Lab……….Thanks you Muhammad and Melanie.

Underground RR/Religious Inst

Old Building



Wayfinding System

National Advertising Campaign

Brooklyn Law School

Historic Landmark


Engineering Marvel/Water

Living Animals