The Mathematician’s Version

Arctic Sea Ice

The National Snow and Ice Data Center (NSDIC) in Boulder, Colorado has been monitoring the extent of the September Arctic sea ice, in millions of square kilometers since 1979.

Data from the NSDIC website are given at left.

Data source: http://nsidc.org/news/press/20050928_trendscontinue.html

Arctic Sea Ice

We let t represent the number of years since 1980and f(t) represent the extent of the September Arcticsea ice, in millions of square kilometers. We use the data points corresponding to the years 1980 and 2000 to construct a linear model. The data points are (0, 7.8) and (20, 6.3). This gives the model

f(t) = -0.075t + 7.8

Arctic Sea Ice

Arctic Sea Ice -- From the Assignment

First body paragraph: Here you should examine your data on extent of September Arctic sea ice and present the mathematical model you have developed. Explain where the data came from, how you developed your model, and (very importantly) what it (the model) definitely tells you. A table of the data and a graph showing the data and model, all carefully labeled, should be included. You should also discuss, qualifying your statements carefully, what the model seems to indicate (but does not prove).

Further body paragraphs: Devote one paragraph to a discussion of the predictions of your mathematical model on the extent of the Arctic sea ice, and discuss what this would mean for the Northwest Passage. In another paragraph discuss whether or not the data and your model are consistent with the hypothesis that global warming is occurring. You will also want to think about the possible limitations of your model. Do you expect it to be an accurate forecast of the future? Or do you think the model might break down in some fashion? If you expect that your model might break down, discuss the factors that you think will cause the model to break down and the ways in which this will affect your predictions. This data, and your model, by themselves, are not enough to prove that global warming is occurring. Discuss what additional information or data you would need in order to to be able to conclude, with certainty, that either global warming is occurring or that global warming is not occurring.

Arctic Sea Ice -- Student Paper

… our linear model:

f(x) = -0.0708x + 7.8

By looking at the graph derived from the data and model points, we predict that the extent of the September Arctic sea ice in the year 2050 will be approximately 3 millions of square kilometers. We can also predict from the graph that the extent of the September Arctic sea ice will reach zero at approximately the year 2090.

Our data and model seems to be consistent with the hypothesis that global warming is occurring. As Earth’s temperature increases, the Arctic sea ice thins out and the extent might possibly reach zero. If global warming were said to not be occurring, the ice extent would not decrease over time. The model, however, has limits. Trends do not always continue in the same direction. There is a possibility that the model might breakdown due to unexpected events such as the Earth’s temperature decreasing, less production of carbon dioxide, or other environmental factors to make our predictions inaccurate. Also, our economy might reduce the burning of fossil fuels with a better energy source or develop a technology where less carbon dioxide and other heat-trapping gases will enter Earth’s atmosphere... Additional, accurate information is required for us to conclude about whether global warming is occurring or not occurring. Information about whether or not carbon dioxide will indeed cause global warming and the amount of warming we should be aware of is needed.

Arctic Sea Ice -- Student Paper

Appendix

We choose data points corresponding to the years 1980 and 2004 which are (0, 7.8) and (24, 6.1). To find the model, we choose ordered pairs (years since 1979, data).

First, we solve for the slope by using the selected data points:

m = (6.1 – 7.8) ¸ (24 - 0) = -1.7 / 24 = -0.0708

Then, we use the slope we found to put into the point-slope equation, using one of the selected data points as x and y to find the model:

y2 - y1 = m(x2 - x1)y – 7.8= -0.0708 (x – 0)

y – 7.8 = -0.0708xy = -0.0708x + 7.8

f(x) = -0.0708x + 7.8- Using the linear model:

Predict the year the extent will reach zero 0 = -0.0708x + 7.8 0.0708x = 7.8 x = 110.169

110.169 is the approximate years since 1979.

We predict that the extent will reach zero approximately in the year 2089.

Arctic Sea Ice -- Student Reflections

I found it difficult to write about why the statements could not be proved. It’s hard to take a passive stance while writing an essay when I have always been taught otherwise.

It was interesting to see factual numbers. It was very challenging to try and decide how I was going to remain sort of unbiased when I think our pollution is causing ecological damage.

The most challenging aspect of writing this paper was putting the mathematical concepts into words that made sense.

The challenging aspect of examining the data was making sure that I didn’t try to prove or disprove anything. I’m use to drawing some sort of definitive conclusion in my essays.

Examining global warming in light of new data is a new idea for me. Using mathematics and recorded data does give an essay more credibility because the sources are incorporated rather than just reference.

What about grading papers . . .

if you’re not an English teacher?

When I grade the papers that the students write, I find, generally, that assigning a grade is a relatively straightforward decision. I use the following criteria:

• does the paper fulfill the assignment?

• is the mathematics in the paper correct?

• is the paper logically sound?

• are the ideas expressed with clarity?

To help, I use a rubric as I grade, and I return the rubric to the student as an explanation of the grade awarded.

Malthus, Exponential Growth, and Carrying Capacity

The Moose of Isle Royale

The Moose of Isle Royale

The Moose of Isle Royale

• Isle Royale is an island in Lake Superior• 45 miles long, 8 miles wide• An isolated location• Two species: wolves and moose• In the early 1980’s, parvovirus decimated the

wolf population, removing the natural check on the moose population

carrying capacity

2,500

2,000

1,500

1,000

500

0

# of

moose

20001995199019851980year

Moose of Isle Royale

data

model

Thomas Malthus

• Malthus published his Essay on the Principle of Population in 1798

Malthus’s Argument

I think I may fairly make two postulata.

First, That food is necessary to the existence of man.

Secondly, That the passion between the sexes is necessary and will remain nearly in its present state.

Malthus’s Argument

Assuming then my postulata as granted, I say that the power of population is indefinitely greater than the power in the earth to produce subsistence for man.

Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of the second.

Note: geometrical ratio means exponential growth arithmetical ratio means linear growth

Malthus’s Argument

By that law of our nature which makes food necessary to the life of man, the effects of these two unequal powers must be kept equal.

This implies a strong and constantly operating check on population from the difficulty of subsistence. This difficulty must fall somewhere and must necessarily be severely felt by a large portion of mankind.

Malthus’s Argument

Premise 1: Population will increase by exponential growth

Premise 2: Subsistence will increase by linear growth

Conclusion: Population will increase beyond the ability of mankind to provide subsistence for the population. The consequences of this will

be “severely felt by a large portion of mankind”.

World Population

Burke: from the World Population Assignment

Initial Mathematical Work:

Before you write your essay, begin with a mathematical analysis. Some initial mathematical work will get you started.

• Construct a mathematical model for the growth of world population. First, look at the graph of the data points, and decide whether a linear or exponential model would be more appropriate. Then define appropriate variables. Select two data points and use these to construct your model.

• Add your model to the spreadsheet, and then graph the data and model. Extend the table and the graph into the future to the year 2050. Compare the graph of the model to the data points. Is your model a good fit to the data? The mathematical details of this work should be presented in the appendix. • Use your model to predict world population in the year 2020, and to predict the year in which the population will reach 10 billion. Mathematical details of these calculations should be shown in the appendix.

• Verify that your calculations are correct by using both the table and the graph to make the same predictions. Be sure to incorporate the answers to the above questions in your discussion of the model in your paper.

Burke: from the World Population Assignment

Your paper should focus on two central issues: when and how will your model break down if we do nothing (as we are now doing), and what should we do to prevent the tragedy of the commons? Of course, your argument should be informed by your work with the data and your projections into the future.

To address the question of model breakdown, you will need to carefully consider carrying capacity. Of course, we do not know the carrying capacity of the earth, but think about what we would need to know to decide on a reasonable carrying capacity. Certainly, the carrying capacity will depend upon both the population level and upon patterns of consumption. Finally, speculate about what you think the carrying capacity might be. What are the implications of your (considered) guess about carrying capacity for the tragedy of the commons?

As you consider a solution to the tragedy of the commons, you might want to examine strategies advocated by some of the essays you have read. Hardin suggests that private property is a solution for some of the problems of the commons. Aldo Leopold, in “The Land Ethic,” advocates an ethical change in the way we see ourselves in the world. … Do you think any of these strategies offer a realistic solution to the tragedy of the commons? Or do you have another strategy to suggest? Or perhaps you think that the tragedy is inevitable. Support your position with a carefully reasoned argument.

Burke: Student Paper

This gave me the equation P(t) = (1.86)e.013755(t) for my model. The algebra used in reaching my model is presented in the appendix to this paper. I have presented the UN’s data along with my model and graph below.

t Year Data Model

-20 1900 1.65 1.4126593

-10 1910 1.75 1.6209708

0 1920 1.86 1.86

10 1930 2.07 2.1342766

20 1940 2.3 2.4489981

30 1950 2.52 2.8101287

40 1960 3.02 3.2245118

50 1970 3.7 3.7

60 1980 4.44 4.245604

70 1990 5.27 4.871663

80 2000 6.06 5.590041

90 2010 6.414351 4

100 2020 7.360215

110 2030 8.445556 2

120 2040 9.690942 4

130 2050 11 .11997 4

140 2060 12.75973 1

W o r l d P o p u l a t i o n T r e n d s

0

2

4

6

8

1 0

1 2

1 4

1 9 0 0 1 9 2 0 1 9 4 0 1 9 6 0 1 9 8 0 2 0 0 0 2 0 2 0 2 0 4 0 2 0 6 0

Y e a r

Population (in billions)

U N D a t a

M o d e l

Burke: Student Paper

Is mutual coercion the answer? Is education the answer? Or, will another short-term answer be found through technological advancements? I can not answer these questions, but I can make my own predictions from the graph. In my opinion, something will happen to level off the population growth. It may be a combination of changes made by man, or it may be forced upon us by nature in the form of food shortages, drought, or disease. I would like to note that Hardin’s predictions were incorrect, for he did not foresee the Green Revolution and it’s effects, and it is very plausible that Erlich’s opinions will also be outdated by similar advancements in genetics, chemistry, and techniques to desalinize ocean water. Every living organism makes adaptations to improve it’s own survival or it becomes extinct. Humans are no different. … Perhaps, it is time our culture had a wake up call, and just maybe, we would find a way to live without constant digital stimulation, commuting sixty or more miles to put in a day’s work, and split second internet connections. After all, the rest of the world has been surviving just fine without Starbucks. Again, nobody truly knows what will happen. We can only use the information we are given to make predictions, and the data definitely shows that drastic changes must be made if we hope to curb the growth rate problem. Until these changes are made, I think the model formulates an easy argument to suggest trouble in this lifetime.

Burke: Student Paper

Appendix

For my model I started with a basic exponential formula: P(t) = (P0)e k(t)

The (t) in the formula represents time in years. The (P0) is used in place of the initial population in billions, and the (k) is a constant.

First, I found two points on the graph that appeared to fit my exponential growth line. I used the first of these points for my initial population, setting (t) at zero for this point and then solved for the second point accordingly:

3.70 = (1.86) e k(50)

I was able to divide 1.86 from each side: 3.7/1.86 = e k(50)

Next, I took the natural log of both sides: ln(3.7/1.86) = k(50)

Finally, I divided by 50 and solved for (k): k = ln(3.7/1.86) / 50 = .013755 Substituting this (k) value we have our model: P(t) = (1.86)e .013755(t)

Recall: Initial Focus

My initial focus was relatively narrow. I saw the data-based assignments as an interesting supplement to my mathematics classes. In particular, I wanted to:

• ask my students to use a spreadsheet to examine functions from numeric, geometric and analytical points of view

• offer my students genuine applications of mathematics

• teach through interdisciplinary problems, so that my students would see that knowledge is not constrained by disciplinary boundaries

• ask my students to write about mathematics because I had the conviction that writing about mathematics would help them to clarify their mathematical thoughts

Shift of Focus

As the year progressed, my focus shifted. I am now more interested in using the assignments as a way to provoke discussion of the following questions:

• how do we decide what is really true?

• can we make decisions for ourselves, or do we have to rely on experts?

• what is the proper role of data and evidence in the making of decisions?

• where do reliable data come from?

• how do we treat and interpret the data?

• what can we properly conclude after we look carefully at the data?

• how can a mathematical model break down?

• what are the implications when a model breaks down?

• what is the interplay between data, preconceptions, opinion, and belief?

Contemporary Issues: the Exercises

Linear Models:

• Global Warming 1 - CO2 Data • Klamath Salmon

Exponential Models:

• Spread of AIDS• Carbon Dating - Libby • Yellowstone Wolves

Breakdown of the Model:

• Population of Ireland• Global Warming 2 -- Arctic Ice Cap

Additional Exercises:

• Olympic Running Times (the Hundred Meters)• Nuclear Waste -- Bethe • World Population -- Hardin

How can this material be used?

• as supplementary assignments for a standard mathematics course

• as the basis for a stand alone mathematics/quantitative reasoning course

• as the basis for a learning community that involves mathematics

• as central problems for a confluence model structure

• as central problems for a freshman experience

-----> discussion on next slide ----->

Ideas

• Lewis and Clark, and the American West-- Klamath Salmon, Global Warming 2 (Arctic Ice Cap), Yellowstone Wolves

• Los Alamos, the Bomb, and Nuclear Power-- Global Warming 1 (CO2 Data), Nuclear Waste (Bethe)

• The Tragedy of the Commons or Collapse-- Klamath Salmon, Population of Ireland, Global Warming 1, Global Warming 2, World Population

• Energy-- Global Warming 1, Global Warming 2, Nuclear Waste

• Wildlife and Ecology-- Klamath Salmon, Moose on Isle Royalle

Ideas That Are Central to My Work

Integrative Learning

Writing Across the Curriculum

Quantitative Literacy

Critical Thinking

Citizenship

Uncertainty

Integrative Learning

Fostering students’ abilities to integrate learning -- across courses, over time, and between campus and community life -- is one of the most important goals and challenges of higher education. The undergraduate experience can be a fragmented landscape of general education courses, preparation to the major, co-curricular activities and “the real world” beyond the campus. But an emphasis on integrative learning can help undergraduates put the pieces together and develop habits of mind that prepare them to make informed judgments in the conduct of personal, professional, and civic life.

Integrative learning comes in many varieties: connecting skills and knowledge from multiple sources and experiences; applying theory to practice in various settings; utilizing diverse and even contradictory points of view; and, understanding issues and positions contextually. Significant knowledge within individual disciplines serves as the foundation, but integrative learning goes beyond academic boundaries. Indeed, integrative experiences often occur as learners address real-world problems, unscripted and sufficiently broad to require multiple areas of knowledge and multiple modes of inquiry, offering multiple solutions and benefiting from multiple perspectives.

a statement on integrative learningassociation of american colleges and universitiesthe carnegie foundation for the advancement of teaching

Integrative Learning

There are a variety of structures that can be used to promote integrative learning:

learning communities

confluence model

core curriculum

first year seminar

freshman experience

new, integrative stand-alone courses

OR even . . .

existing, stand-alone, gen-ed courses

Writing Across the Curriculum (or Written Communication )

Writing is too important to be left to the composition faculty alone.

“Writing professionals no longer view composition as a mechanical processof turning previously formed ideas into suitable prose but as something inseparable from thinking itself. . . . Only through writing, perhaps . . . through the writing of drafts of papers that required them to integrate theory with evidence, did [students] achieve the insights that moved them to complex reasoning about the topic under consideration.”1

Derek Bok

“ -- we write to figure out what we think -- we write to to learn how to think -- mathematicians have been largely absent from this effort -- I want to provide examples of ways in which we can participate”

Michael Burke

1.In Our Underachieving Colleges by Derek Bok, p.92

Quantitative Literacy

Quantitative Literacy is too important to be left to mathematics faculty alone.

“My thesis today is that by virtue of our training, mathematicians have distinctive habits of mind that can enhance public discussion of public issues. More importantly, we have a professional obligation to move beyond the boundaries of our own discipline to bring our special skills of analysis and clarification to bear on important public policy discussions.”1

Lynn Steen

Quantitative literacy is “intertwined with political, scientific, historical or artistic contexts. Here QL adds a crucial dimension of rigor and thoughtfulness to many of the issues commonly addressed in undergraduate education. … QL is not a discipline but a literacy, not a set of skills but a habit of mind.” 2

Lynn Steen

“Numbers: it’s where it’s at.”

Michael Burke

•“On Being a Mathematical Citizen: The Natural NExT Step” by Lynn Arthur Steen in Focus Magazine, October, 2007

• In Achieving Quantitative Literacy: An Urgent Challenge for Higher Education, by Lynn Arthur Steen. p 22

Critical Thinking

Critical thinking is too important not to be taught in every discipline.

“An intriguing question or problem is the first of five essential elements that make up the natural critical learning environment.”

Ken Bain“ -- the distinction I see is between thinking and non-thinking

-- wishful thinking -- reflexive thinking-- magical thinking -- dutiful thinking

-- I am interested in teaching elementary scientific thinking that:-- relies on data and evidence-- requires a willingness to question, to conjecture, and to test conjectures against the evidence-- has mathematical reasoning at its core

-- the exercises are designed to:

-- demystify science -- teach scientific thinking-- to demonstrate to our students that they can think like a scientist-- to illustrate the power of scientific reasoning”

Michael Burke In What the Best College Teachers Do by Ken Bain. p 100

Citizenship

Uncertainty

Overview

1. Introduction

2. Exercise: Klamath Salmon

3. Overview

4. Exercise: George Will, Al Gore, Arctic Sea Ice

5. Exercise: Moose, Malthus, World Population

6. Thoughts: Using This Material

7. Thoughts: Ideas that Are Central to My Work

8. Conclusion

A Faculty Member’s Reflections

In the learning community, requiring students to grapple with data and modeling in their papers made them aware of their obligation to consider facts and logical thinking in addition to their "feelings." When I gave students in other classes the same assignments, but without the data, the papers seemed easier for the students to write but lacked evidence of the healthy intellectual struggle that resulted from confronting the numbers.

Jean MachProfessor of EnglishCollege of San Mateo

One Student’s Reflections over the Year

I did not find examining the statements and how they compared to the data particularly challenging. As mentioned in the paper, I’ve found no conclusive evidence (and do not want to jump to conclusions) that manmade CO2 emissions had any effect on global temperatures. Indeed, the data made no mention of global temperature trends. Even if CO2 levels and temperatures rose during the same time period, correlation does not imply causation. There is no conclusive evidence that such emissions (which may or may not be manmade) have any significant effect on temperature.

I felt as if I did have a preconceived idea about a conclusion. Specifically, that nuclear power is a safe, effective means of power generation. Still, I approached the issue with an open mind and tried to be objective with all the data given. Although I reached the conclusion that nuclear power is safe, I was able to appreciate the concerns of those who oppose it.

The idea of dealing directly with the numbers and building one’s thoughts and opinions of an issue off one’s own analysis (rather than reading someone elses attempts at persuasion) is a much more logical approach that minimized bias and error. Thinking about and analyzing data this way was very productive.

Concluding Thoughts -- Student

Taking data from one source and having that data predict what the situation where like if 50 years were to pass are something that I have learned from this paper. Also just because a data model predicts that something is going to happen does not mean that that particular event is going to happen because there are too many unknowns that happen everyday and those unknowns might alter the situation and change the trend.

However, the most important item that I have learned from this assignment was not how to make a graph with the computer,or crunching out numbers from handouts given to us. But it was making sense of the data which was handed to me and piecing together those data to make a solid thought.

fin

Additional ResourcesIntegrative Learning Initiatives at College of San Mateo:

http://www.collegeofsanmateo.edu/integrativelearning/

Tools for Thought Learning Community: http://www.smccd.net/accounts/csmlcom/ToolsforThought/tools.htm

Books about Quantitative Literacy:

Mathematics and Democracy: The Case for Quantitative LiteracyLynn Arthur Steen (Editor)

Achieving Quantitative Literacy: An Urgent Challenge for Higher Education Lynn Arthur Steen (Editor)

More reading about Mike’s work:

Carnegie Perspectives Piece:

http://www.carnegiefoundation.org/perspectives/

Carnegie Scholar Final Project Snapshot:

http://www.cfkeep.org/html/snapshot.php?id=61548260563564