Two Perspectives on Integrative Learning and Quantitative Reasoning
Michael C. BurkeCollege of San Mateo
Innovative Pedagogy and Course Redesign IXFairfield University
June 4, 2009
AAC&U: The Essential Learning Outcomes
• Knowledge of Human Cultures and the Physical and Natural World
- Through study in the sciences and mathematics, social sciences, humanities, histories, languages and the arts
Focused by engagement with big questions, both contemporary and enduring
• Intellectual and Practical Skills, including - Inquiry and analysis - Critical and creative thinking
- Written and oral communication - Quantitative literacy
- Information literacy - Teamwork and problem solving
Practiced extensively, across the curriculum in the context of progressively more challenging problems,
projects, and standards of performance
• Integrative Learning, including
- Synthesis and advanced accomplishment across general and specialized studies
Demonstrated through the application of knowledge, skills, and responsibilities to new settings and
complex problems
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
“Quantitative literacy is more a habit of mind, an approach to problems that employs and enhances both statistics and mathematics … Unlike mathematics, which is primarily about a Platonic realm of abstract structures, numeracy is often anchored in data derived from and attached to the empirical world.” 1
The Quantitative Literacy Design Team
Quantitative literacy is “about challenging college-level settings in which quantitative analysis 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
The Quantitative Literacy Design Team
Richardson and McCallum argue that “Quantitative literacy cannot be taught by mathematics teachers alone, not because of deficiencies in teaching but because quantitative material must be pervasive in all areas of students’ education .” 3 For this reason the writing-across-the-curriculum model seems to offer a promising approach.
1. In Mathematics and Democracy: The Case for Quantitative Literacy, edited by Lynn Arthur Steen. p 5.
2. In Achieving Quantitative Literacy: An Urgent Challenge for Higher Education, by Lynn Arthur Steen. p 22
3. In Achieving Quantitative Literacy: An Urgent Challenge for Higher Education, by Lynn Arthur Steen. p 39.
WACPack
The Moose of Isle Royale
QuickTime™ and aTIFF (Uncompressed) decompressor
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carrying capacity
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carrying capacity?carrying capacity?carrying capacity?
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I invite you to consider the following question:
What is the carrying capacity of the earth?
• Is it 10 billion, so that we have not yet reached it?
• Is it 6 billion, so that we are passing it about now?
• Is it 3 billion, so that we have blown past it, like the Isle Royale moose?
Mach Assignment
Mach: from the World Population Assignment
You have examined data on the growth of moose and human populations by graphing them, identifying trendlines, and making projections for the future. You have given some thought to the implications of your projections. You have also read the article “Optimum Human Population Size,” by Gretchen C. Daily and Anne H. Ehrlich and Paul R. Ehrlich, which gives a summary of the authors’ opinions on the issue of world population. As good critical thinkers, you should remain open to new information as it tests the hypotheses that are forming in your mind. In this case, you should be making connections between these two sources of information, comparing your ideas with those in the reading and trying to reconcile them. In what ways are your WACPack answers consistent with the opinions in the reading? In what ways are they not? Would you change the opinions you were formulating in light on this new information or not?
This essay should be “expository,” explaining for the reader the thinking processes you are going through in integrating and reconciling the information you were given. It is not meant to be an “argument” or persuasive piece of writing, although the thinking that you are probing and revealing—the comparing, the evaluating, and the development of new hypotheses—is the basis for good argument.
Mach: Student Paper
In the first section of the WACPack, I looked at a preserved small isolated ecosystem of Isle Island that primarily consisted of moose and wolves. The moose population was kept at approximately 1000 due to their predator, the wolf. However, in the 1980’s the wolf population was hit with a fatal virus, killing off the majority of wolves. The impact on the simple ecosystem was dramatically seen throughout the years of an increase in the moose population. I was given numbers of the population size from 1980 to 1995 and was asked to plot the data. Next I was asked to predict what would happen in the years after 1995. Would the population continue to increase? Or would it decline dramatically? Initially I only took into consideration the carrying capacity of the island. I figured that the population would grow until the land was completely covered, topping at 4,200 moose in the year 2002. However, the data revealed that in the next year, 1996, the moose population actually dropped from 2,422 to 1,163 and continued to decline until about 2002 were it leveled back at 1000. This shows that the optimum population size for the land, regardless of natural predators is about 1000 moose. I feel that my predictions were too high primarily due to the fact that I did not consider the ‘lifestyles’ of the moose. I found that putting a term such as ‘quality of life’ on wild animals was unheard of. However, it seems that even animals maintain a certain quality of life and by doing so it affects the population trend of the species.
Mach: Student Paper
Next, I looked at the trend of human populations versus moose populations. In the example of the Isle Island, the moose and wolves exist on a very simple ecosystem, where moose are the prey and the wolves are the predator. This I believe allowed for the natural progression of nature to keep equilibrium. I do not see this in the human population due to technological inventions and medical achievements altering the course of nature. Humans are directly altering the natural outcome of life by changing nature with things such as cloning, synthetic medications and genetically engineered foods. However, there are only a few things left that level out or acting as a ‘predator’ to humans these are natural disasters, super viruses and war. I feel that these predators are the only way humans will self regulate and find equilibrium. However, I do not feel that natural disasters, super viruses, or wars are in abundance to combat the exponential growth rate of the world, especially in developed countries that have such advanced technology and weaponry to fight these natural killers.
Mach: Student Paper
When I read the article, ‘Tragedy of the Commons’ by Garrett Hardin, I was impressed by his argument against overpopulation. Hardin brings up the same issues that are addressed in the WACPack, that it is impossible to have two things at their maximum, such as population and a high quality of life. In the OHPS, it states that we cannot have a large population because the earth cannot provide enough resources to maintain a high standard of life to all. Our world now can provide a high quality of life to only a small portion of the population, primarily ones in developed countries. In addition, the small populations with the higher quality of life attribute to the majority of the waste produced on the earth. This imbalance is one of the reasons why the author of the ‘Tragedy of the Commons’ believes we must address this issue now to prevent to destruction of our species.
Throughout my journey of thought I have uncovered many revelations about the population problem our world faces. I have read three articles that confirm and contradicted some of my original beliefs. I now truly feel that our population will act as the moose population did on Isle Island and decline once we hit our maximum carrying capacity. However, the quality of life for many people will decline as we start to reach our maximum. I do not know or find any supporting data that gives me any hope that we will achieve our optimum population size. At some point our planet will reach its tipping point and start to decline by some unknown force; whether it is by war or disease, only the future will tell.
Burke Assignments
Burke Assignments
For the past few years, I have been designing, and assigning, data-based integrative writing assignments in my mathematics classes.
Each assignment presents the students with a data set about an important issue. Students are asked to analyze the data mathematically by constructing a mathematical model, and then to use a spreadsheet to implement the model.
They are to produce a written paper in which they present their model (with a table and a graph), and then to use this work as a basis for any conclusions that they reach. They are to show the mathematical details of their work in an appendix to the paper.
Burke Assignments
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
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
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
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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)
Burke: Student Reflections
I think writing about math really helps you understand or forces you to understand what you are doing. The integration of mathematical modeling and writing, I think, has introduced me and taught me about how to support my arguments logically using data. My writing is much more structured and thorough in addressing all questions.
Graphing has always been my mathematical Achilles heel. But, with the papers we’ve written, I’ve come to appreciate graphs. Sometimes they might be hard to read, but they do give such a clear visualization of data. I like mathematical modeling because it is open-ended -- unlike most math, there is no right or wrong answer when interpreting the model’s projections.
The thing I enjoyed about writing mathematical papers was being able to apply mathematics to real life scenarios. It has positively affected my writing by merging English writing and Math. It helped my understanding because I have something to relate math to in a real world circumstance.
My impression before studying this subject is that the earth could easily sustain double its current population. Wow, am I off base. Although the mathematics was interesting, necessary, and fun, the papers made this class special. This course was the most important and most fun I’ve had in any college course.
Burke Assignments
As I worked with these assignments, 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?
What about grading papers?
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.
Background Readings for Both Assignments
“The Tragedy of the Commons”, in Science, by Garrett Hardin, 1968
“Hostages to Hubris” from One with Nineveh, by Paul and Anne Ehrlich, 2004
“Optimum Human Population Size”, in Population and Environment: A Journal of Interdisciplinary Studies, by Gretchen Daily, Anne Ehrlich, and Paul Ehrlich, 1994
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