Quantitative Reasoning

Core Biology 8.1

John Blamire

Introduction

We all see the world in different ways, from different perspectives and with a wide range of points of view. Scientists are supposed to have yet another way of looking at the world; they are challenged to look beneath the surface of a phenomenon and discover the underlying mechanism whereby it works. One of the many tools they use to accomplish this task is mathematics and a skill in quantitative reasoning.

Beyond the ability to perform simple calculations, scientists must be able to use mathematics to "reason" and reveal hidden patterns or trends that explain why a particular action results in a totally predictable outcome. In Core Biology 8.1 we expose undergraduate students not only to what is known about the mechanisms that operate within living organisms, but also to why it is necessary to use mathematical reasoning to understand how we know about these mechanisms.

Core Biology 8.1 contains traditional content, active learning, and hands on exercises in which students use simulations to explore biological phenomena, and then apply the principles of quantitative reasoning to explain what they have found.

A Case Study: the discovery of genetics.

The invention of the microscope put a new and valuable tool into the hands of Robert Hooke and his friends. Using the power of magnification, they were able to see things that had previously been unimagined: the world of cells. The branch of science we now call "genetics" was discovered in a different way.

Every year millions of pea plants had been grown and harvested by farmers and gardeners around the world. These growers had all seen differences in flower color, pod shape, height and many other characteristic features inherited from one generation of peas to the next, but had never been able to answer the simple question, how does this happen?

In the 1860's Gregor Mendel studied the patterns of inheritance as seen in the common, garden pea plant and saw what all those other growers had seen. For example, the color of peas could be green or yellow, and this property was clearly passed on (i.e., inherited) from parent to child to grandchildren. Unlike all the other growers, however, Mendel asked himself the question, "was there any reasonable underlying mechanism to this pattern of inheritance, or was it all random and unexplainable?"

The answer, as Mendel showed, lay in the numbers. One crop of 582 plants gave 436 with yellow seeds and only 146 with green seeds. The next crop of 239 plants gave 179 with yellow seeds and only 60 with green seeds. In the next crop of 422 plants, 317 had yellow seeds and 105 had green seeds.

Somewhere in these numbers, Mendel reasoned, lay an explanation of what was going on. Looking at the raw data gave no clue. What did it matter if the crop size (436, 239 and 422) was different every year? What is significant is that there always seemed to be more yellow seeds than green seeds, but did this say anything about how the property was being inherited? Possibly, but what?

Fortunately, Mendel had been to University (not Brooklyn College, but somewhere almost as good!), and had studied some physics. In this new branch of science, scientists were constantly uncovering fundamental principles by manipulating the numbers they got when pendulums swung back and forth or electricity moved along wires. It was an exciting idea.

For hours, Mendel stared at his raw data. The simple numbers showed no clear pattern; it seemed that more plants inherited the ability to produce yellow seeds than green seeds, but every year the number of plants varied unpredictably and the raw numbers had no particular pattern to them.

So Mendel turned the raw data into simple mathematical ratios. In the first crop the ratio of yellow seeds (436, raw data) to green seeds (146, raw data) was 436/146 almost 3:1, as was the next ratio in the set of data (179/60), and the next, and the next and the next!

Suddenly, there was a consistency; the number of plants producing yellow seeds was always three times greater than those producing green seeds. Surely this meant something!

The ratio of 3 yellow to 1 green was a fundamental property of this type of genetic cross, and the inheritance of seed color had to be explained by a mechanism that accounted for the consistency of this ratio. There WAS a mechanism of inheritance, and it worked by the numbers!

Using a different kind of reasoning, Mendel proposed a mechanism of inheritance that hypothesized a controlling, particulate "elemente" (what we would call a gene today) that produced either the "yellow seed trait" or the "green seed trait". Each parent plant had two of these "elementes" which could be in one of two forms (i.e. "yellow producing" or "green producing").

He went on to propose that, during the process of plant reproduction, every generation of new plants inherited one of these "elementes" from the male parent and another one from the female parent. These two simple hypothesizes (a "gene" and "one from each parent") could completely explain the 3:1 ratio of inheritance, and provided the basis of a lot more experiments and discovery.

The rest, as they say, is history. Mendel's work gave us the first clue as to how genes could be inherited, and once this breakthrough had been made, the next 150 years of research has filled in the blanks. However, it was knowing and understanding how the numbers could reveal secrets in biology that started it all. Numeracy WAS important.

Numeracy and Active Learning

In Core Biology 8.1 many of the fundamental mechanisms and principles that apply to all living things, including ourselves, are explained, discussed, and most importantly, re-discovered by individual students using active learning principles that heavily depend on the skills of numeracy.

Students are asked to explore many of the basic principles of biology using a combination of wet laboratory experiments and/or online computer simulations. In each of these exercises, termed "research investigations", there are several elements: reading background literature, a problem or a questions that needs an answer, the "tools of the trade", which are usually internet delivered computer simulations of the phenomenon, and a method of exploring the data using formulae, graphs, charts, or ratios.

For example, on the "Science at a Distance" web site used by Core Biology 8.1 (see below for URL), there is a simulation that duplicates the very genetic crosses first carried out by Gregor Mendel. Here the students are directed to read some of the background relevant to Mendel, information about peas, the context of Mendel's work and even a fictionalized story. They are then directed to a series of questions that lead them though the principles of diploid genetics, and use a simulation to carry out for themselves crosses between various plants.

The raw data they find is recorded, and then in each case students are asked to explain their raw data using simple calculations of the appropriate ratios. The explanation they "invent" for themselves (which explains the mechanism of inheritance) must then be justified by its ability to re-create the ratios they found from the simulation.

In this way, students discover the "secret" mechanisms that are important in biology by the active process of generating and explaining data - not simply by reading it in a book. This is important. It has been shown over and over again that active learning reinforces the material learned and makes it easier to remember and understand.

There are many other examples of active learning exercises based on computer simulations on the Science at a Distance web site, and various instructors in Core Biology 8.1 use and adapt them to their needs and style. However, in almost all cases, the underpinnings of the work done by the students in these exercises (and what they discover) are firmly grounded in mathematical reasoning.

The Future

We intend to continue developing more and more interactive research investigations that use these principles and to create a whole introductory biology course that can be delivered this way. We hope that such a course will not only teach students some biology, but also encourage them to both do more writing and explore the world of mathematics.

Numeracy skills put into the hands of our students are a tool just as important for them as the microscope was for the 17th century scientist. So many of the operating mechanisms that underlie our everyday world can only be seen, explored and understood using the tools of mathematical reasoning. Without these skills, our students are turned out into the world intellectually blind. They can see no deeper than the surface and understand no more than they can see with their eyes.

Hopefully, sometime in the future, when all memory of Core classes at Brooklyn College has faded from their minds, they will still remember that an explanation for any mystery in their lives can probably be revealed to them, as it was to Mendel 150 years ago, by using the numbers.

Let's hope so.

Web-sites:

Science at a Distance:

http://www.brooklyn.cuny.edu/bc/ahp

mechanisms of inheritance:

http://www.brooklyn.cuny.edu/bc/ahp/MGInv/MGI.Inv.html

the Curse of Amun:

http://www.brooklyn.cuny.edu/bc/ahp/AVC/MacroInv/DNA/VCB_DNA_HP.html

Brother Gregory's dinner:

http://www.brooklyn.cuny.edu/bc/ahp/AVC/VCB_BE_HP.html