Consortium for Mathematics and its Applications


The BioMath Connection (BMC) & Integrating Mathematics and Biology (IMB) Projects

BMC/IMB was a pioneering project linking biology and mathematics in the high schools. It provided an opportunity for high school teachers, writers, researchers, and others to get in on the ground floor of developing innovative classroom materials. The materials consist of 20 modules that can be flexibly adapted for use in a variety of courses at a variety of grade levels in both biology and mathematics. The project was run by DIMACS at Rutgers University in collaboration with the Consortium for Mathematics and its Applications (COMAP) and Colorado State University (CSU).


Biostatistics in Practice: Using Statistics To Discover Links Between Eating Behaviors And Overweight Children

Biostatistics (a contraction of the words biology and statistics) is the application of statistics to topics in biology. This unit deals with two biological contexts. In Lesson 1 students look for factors that help them estimate a person’s age. They use dotplots and numeric summaries for center and spread to analyze data from team age estimates. Lessons 2 - 4 are based on the Infant Growth Study that sought connections between eating behaviors in children and being overweight. Students grapple with determining which factors are the best indicators that a child is overweight. Using scatterplots and regression lines, students study relationships between various factors (BMI at age 4, rate of eating during test meal, calories consumed during test meal) and BMI at age six. After creating binary variables (overweight (yes/no), exceeded recommended dinner Calories at test meal (yes/no)), students again search for relationships between factors and the variable overweight. This time the statistical analysis is based on two-way tables and calculating percentages.

Competition in Disease: Pass It On!

Members of a population display many variations. They may be visible traits or internal differences in genetic make up. These variations may result in either “success” or “failure” of the individuals. The concept of natural selection is developed in that those organisms with more successful traits reproduce at a greater rate than those with less successful traits. Some research uses the concept of infectious disease-causing organisms to model this concept. The ability to increase their rate of transmission from one host to another increases these organisms’ chance of being successful in an evolutionary sense; therefore, various methods of transmission are important to discuss. Mathematical analysis of these different methods of transmission provides insight into their possible evolutionary success rates and perhaps even that of the organisms that utilize these methods. This analysis illustrates connections between reproductive success and evolutionary success.

CRIME: Criminal Investigation through Mathematical Examination

Welcome to our unit on the mathematical examination of fingerprints! This unit is provides a possible method to identify individuals in a species. Tracking individuals can be an important step in learning more about the species as a whole. This unit concentrates on human fingerprints. To accomplish this, we use a fictitious crime case to explore a mathematical procedure to examine and compare fingerprints. The unit consists of multiple activities to investigate the characteristics of a fingerprint and the biology behind why everyone’s fingerprints differ.

DNA Sequencing and Sorting: Identifying Genetic Variations

Each of the cells in your body contains a copy of your genetic inheritance, the DNA that has been passed down to you, one half from your biological mother and one half from your biological father. This DNA, mixed and mutated in your ancestors over the generations, determines in large part, obvious physical features like eye color and hair color, and, combined with environmental factors like diet and exposure to toxins, can determine susceptibility to medical conditions like hypertension, heart disease, diabetes, and cancer. In this unit you will learn about the joint laboratory and computer technique called genome sequencing, which is being used to study DNA and identify the variations that exist among people. Variation detection is important for understanding how our genomes differ, which will lead to a better understanding of human health and disease.

Drawing Lines: Spatial Arrangements of Biological Phenomena

One of the fundamental needs of any organism is space in which to exist. Depending on how organisms engage in vital activities, such as finding shelter, foraging for food, courtship, and reproduction, the spatial needs of different species can vary and interact. Some animals range over many hundreds of square miles while foraging, while others never leave the small pond in which they were born. Many animals, regardless of their sizes or the scale of their habitats, are very territorial. These territories can be formed both within and between species. In addition, there are many other scales at which the biological organization of space is important. For example, at the cellular level, some cells interact with their neighboring cells, and at the ecosystem level all of the plants that comprise a forest together structure the space in which they grow.

Ecological Footprint: What’s My Impact?

Most people think of humans as being separate from the environment and when they think about ecology it is usually the ecology of some nonhuman ecosystemHabitat FT Visuals.ppt themselves, and humans in general, as intimately connected to the environment. It introduces the ecology of humans as a topic in its own right. This focuses on both the human dependence on the environment and the human impact on the environment’s ongoing capacity to meet the needs of our world’s people. To better understand human impact on the environment, basic mathematics is used to quantify ecological impact. Ecological footprinting is developed as a tool for assessing human impact and as a decision-making tool. This unit deepens students’ awareness of the human role in environmental crises, and enables them to make more informed decisions about their behaviors and their environmental impacts. Although the mathematics is basic, students gain profound insights from consideration of simple ratios and proportions and working with conversion factors. Finally, their work with the ecology of humans prepares them to understand similar issues in other ecologies, and for connecting humans with the environment. The central concept of footprinting is a simple but surprising one with profound implications for human society.

Evolution By Substitution: Amino Acid Changes Over Time

This unit introduces students to the Central Dogma of Biology. In short, DNA, or deoxyribonucleic acid, carries the code for life and that code directs the making of proteins that will carry out the organism’s functions. Proteins are made from twenty different amino acids and the number and order of those amino acids will determine the properties and function of the protein. Any alterations in the sequence of amino acids may have an effect on the function of the protein. The protein may not function as well, may lose all function, or may possibly function better. It is also possible that the substitution may not affect the function of the protein at all. Mathematical analysis of similar proteins in different organisms based on the sequence of amino acids may give insight into their possible evolutionary history and perhaps even that of the organisms that contain those proteins. Such analysis may also lead to explanations of the mechanisms of evolution, which resulted in the natural selection of these proteins.

Evolutionary Game Theory: The Game of Life

This unit examines the role that behavior plays in evolutionary fitness. Through studying and playing games, students will develop an understanding of natural selection as organisms compete for limiting resources (e.g. food, water, space, mates, safety, etc.). Traditionally, this idea is developed in the high school curriculum by focusing on adaptations due to the physical phenotype of the organisms. This unit will look specifically at behavioral choices made to obtain the resources organisms need for survival and for reproduction. The possible choices in this unit have been termed "Moderate" or "Greedy." Moderate means that the player will take only the resources that can be attained without confronting the other player for control of the resources. Greedy indicates the player will confront the other player for control of the resource even if the resource is not immediately needed. By playing the game for M & M's, students will develop different strategies under different conditions and examine the effects of these strategies on survival. If this behavioral trait is due to a genetically inherited variation, then these genes may be passed to the next generation. Those that are more successful will survive to live another day and perhaps even go on to reproduce. Unsuccessful individuals will not reproduce and will die before passing their genes on to the next generation. The successful individuals will pass their genes on to their descendants; therefore, having only the genes of successful individuals represented in the next generation. Building on all of this, the concept of evolutionary fitness is being both successful at surviving and successful at reproducing.

Food Webs: Community Feeding Relationships

Food webs are abstract representations of feeding relationships in communities and use a series of arrows from one species to another where the first is a source of food for the second. Discrete mathematics provides a model for a food web using a directed graph (digraph) whose vertices are the species and an arc goes from a to b if a is food for b. Digraphs representing food webs make understanding predator prey relationships easier and various properties of digraphs provide insight into properties of the food web and the species contained within. Overarching questions in this module include, “What effect would the removal of a species have on the associated food web?” and “Why are there so few top predators?”

Genetic Inversions: Relationships Among Species

The surface of our planet is populated by living things—curious, intricately organized chemical factories that take in matter from their surroundings and use these raw materials to generate copies of themselves. The living organisms appear extraordinarily diverse in almost every way. What could be more different than a tiger and a piece of seaweed, or a bacterium and a tree? Yet our ancestors, knowing nothing of cells or DNA, saw that all these things had something in common. They called that something “life,” marveled at it, struggled to define it, and despaired of explaining what it was or how it worked in terms that relate to nonliving matter.

Habitat: An Organism’s Environment

As human populations increase and spread into areas shared with other organisms, more and more species are added to the endangered species list. As awareness increases about the impact of human activities on the environment, many questions are asked. Are there ways to lessen the negative impact that humans have on other organisms? Can human developments be designed to prevent the demise of other populations of organisms? If, for example, a community feels a golf course would be a nice addition, could it be built to limit its negative impact on other species or even possibly have a positive impact on the environment? In order to answer these questions, it is necessary first to understand the problem at a deeper biological level. Then, mathematical tools can be used to state the problem in precise language and to help one to arrive at a satisfactory solution.

Home Range: Species’ Living Rooms

How do researchers determine the home range of a particular species? What is meant by a species’ home range? How does the home range of a species connect to its habitat? This unit explores how data is collected and analyzed to determine the home range of a number of species. Students use actual data for prairie dogs, black-footed ferrets, and pronghorn antelopes. They determine the home range of these animals, including the size and breadth of the home range, and how one would create a buffer zone for the home range. Students are encouraged to draw conclusions as they compare their data to other student’s data. They consider the usability and the effectiveness of different tracking techniques.

Imperfect Testing: Breast Cancer Case Study

The results of a mammogram, like those of many tests, are not always correct. A false positive test result may create unnecessary anxiety, while a false negative test result may lead to a false sense of security. In this unit, students examine the case of an adult female who learns her mammography test is positive. They then use real data to calculate the probabilities of receiving true (or false) test results and discuss the possible implications of a positive test result, given the properties of the test. These properties, which include sensitivity and specificity, can be used to help determine the rates of incorrect test results. Students also investigate the importance of disease prevalence.

Mathematical Modeling of Disease Outbreak

While the 20th century saw a marked decline in infectious disease deaths and an impressive eradication of some infectious diseases, current populations are still faced with outbreaks of new diseases and the resurgence of some previous declining diseases. Disease control in the 20th century resulted from improved sanitation and hygiene, the discovery of antibiotics and the introduction of worldwide childhood vaccination programs. Science and technology played major roles in these improvements. In the 21st century, scientists, researchers, public health officials and governments continue efforts to control infectious diseases such as HIV, West Nile virus, various strains of influenza, severe respiratory syndrome (SARS) and encephalopathy. This unit is about the study of infectious diseases — their causes, prevention, spread, and control.

Microarrays: Arrays of Hope

In all living organisms, DNA carries the code for life. Your DNA and the DNA of every organism is made up of four basic subunits that are arranged in different orders to provide the secret code of life. This is very similar to the way our alphabet works. Read the following two sentences and visualize in your mind what each of these looks like.
Only one letter in our code was changed, but the meaning of the sentence has changed. The same is true of our DNA. In DNA our alphabet consists only of T, A, C, and G, which represent the four basic subunits of DNA called bases. The bases follow very specific pairing rules to form a molecule of DNA that has 2 strands and resembles a ladder.

Modeling Neuron Networks: The Neuroscience of Pain

The human nervous system is made up of 100 to 200 billion neurons, supported by 1 to 2 trillion glial cells. Each neuron has an average of about 5 thousand synapses. Each of these synapses may release multiple neurotransmitters across a range of concentrations. It’s been estimated that the number of potential neural states that these synapses can produce in a single person is more than the number of molecules in the known universe!

Quorum Sensing: Organisms Communicating and Coordinating

We live in an ever-changing world. Many people crave information about those changes. As a result, new means of communicating are continually evolving. People originally relied on word of mouth. With the invention of the printer, newspapers spread information to more people, more quickly. Later, telephones, television, cell phones and the Internet increased both the rate at which people could communicate and the number of people that the information reached. For many young people, it is hard to imagine a world without online social networks, cell phone texting, emails and television.

Spider Silk: Examining Biological Sequences

This unit asks students to apply knowledge of protein structure and function to pose and answer the fundamental question investigated in this unit, “What alignment of two sequences is biologically most meaningful?” The unit introduces students to the rapidly emerging field of bioinformatics by developing the basic mathematical principles that underlie computer programs used to align amino acids. It also provides instruction on how to use one such resource, Biology Student Workbench. In effect, the students become researchers and should begin to understand how mathematical modeling, computing, and biology can work together to answer important scientific questions.

Tomography: Looking Inside With Outside Tools

Tomography is a form of imaging using sections. Tomography truly brings together mathematics, biology and technology to solve problems in many areas. The unit starts with an introduction to tomography and then introduces some basic mathematics used in computed tomography (CT scans). The unit looks at the biology of the human body, what anatomical conditions can be observed using various kinds of scans, and the differences between CT scans and positron emission tomography (PET scans). Many of you have heard of (or even have had) an MRI or CT scan, or even a PET scan, thus you may find the topic of tomography very interesting.

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This material is based in part upon work supported by the National Science Foundation under Grant Number NSF DRL-1020166. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


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