Objectives
Writing in 1945, the eminent mathematics professor G. Polya had a prophetic vision of the study of mathematics as much more than an esoteric interest of academics.
The future mathematician should be a clever problem-solver; but to be a clever problem-solver is not enough. In due time, he should solve significant mathematical problems; and first he should find out for which kind of problems his native gift is particularly suited. 7
I particularly like Polya's choice of words in referring to math skills as a "native gift," because I most certainly believe that the "kind of problems" that these skills are suited for are those of the environment that we are creating for ourselves to live in. This will become more relevant with each passing year, as, according to the most sophisticated climatologist predictions, human influences on weather patterns will surpass all other natural influences by the year 2050. 8
Another guiding principle from Polya is, "The teacher can make the problem interesting by making it concrete." There may be an unequalled educational opportunity to make math interesting in the climate change issues that we are facing. In fact, it may very well be, through our concerted efforts, that the popular focus on climate change and its myriad consequences can be rendered into a vehicle for inspiring our students to achieve in mathematics. The guiding principle and foundation for each objective I will put forward below is that, by implementing concrete context in math instruction. I expect to encourage students to gain a greater understanding of how mathematics applies to their environment and to become agents of change for their communities, present and future, through their application of mathematical tools.
Not every useful application of mathematics in a student's life will necessitate ecological topics. The application of basic math skills to their personal finances and basic economics, as well as other career fields, whatever they may be, is obviously extremely valid. Assuming the basic mathematical operation and notation skill sets are in place for students, however, the real reason to teach them the advanced fields of algebra, geometry, and above is to prepare them to manage bigger conceptual constructs and to apply problem solving skills in complex scenarios. By giving them the contextual experience of valid, applied mathematics we can not only prepare students for future "out of the box" performance in a more effective and more rigorous way, but we can improve their focus and engagement throughout the course work, and thus their attainment and retention throughout their academic mathematics experiences. The best of contexts will be focused on the issues that are most poignant and, as I make the case above, there are few concerns that could be considered more all encompassing for this generation than the environmental issues that comprise climate change.
I will use four major objectives, each based on the expectation that it can be achieved by bringing environmental contexts to mathematics instruction. These goals are to: 1) Engage students in a sense of urgency about their achievement in mathematics; 2) Improve students' application of mathematics skills; 3) Improve students' decisions about which mathematics skills to apply; and 4) Improve students' abilities to apply skills in pragmatic ways to solve unique, unscripted problems.
The first point, of engaging students, goes beyond the obvious need to engage students in the class work of their mathematics course. I expect that by providing environmental contexts it will be possible to instill a new sense of urgency in their process of thinking about mathematics and its utility as a tool for dealing with climate change. This is a rather large order to fill in a time when the decline and inadequacy of mathematics education in the United States is widely acknowledged and despaired. While every student will not wish to become an environmental scientist, the cohort of adolescents, who are typically of quite a diverse class in the Pittsburgh Public schools, have a strong general aversion to the idea of being a victim of ignorance. This may not be readily apparent in the aversion to learning mathematics among these students, but that reluctance to invest their thought and energy is a product of devaluation of the course content. By applying their study of mathematics skills toward solving issues of environmental consequences and human induced climate change in accessible, observable ways - these same students can be won over to recognition of and investment in their course work. I expect that this investment will extend to students culturing esteem for mathematics as they become more versed in its uses, by further means of environmental context.
An appropriate critique of my expectations as stated here might be, "Do we really expect students to become significantly interested in the minutiae of ecology?" I have been trying to honestly work this question out for myself, whenever this reasonable doubt creeps up. I think, that in the midst of this doubt, both my skeptical colleagues and I are stuck again with that earlier, vague image of a melting ice cap and very little specifics or tangibility about climate change. This bleak landscape, while evocative, isn't necessarily one that rouses a person to action. We might advance the mental picture by introducing a polar bear, adding a factor of cuteness and humane perhaps, but will students be able to connect with a creature that lives far away and might eat them?
I believe that they can actually, though there may be some steps between students' initial awareness of climate change and their initiation into the conscientious stewards of their world. These steps to environmental enlightenment might be better trod toward an animal that lived in the other direction though, to a Central American rainforest amphibian that no longer exists. The golden toad was the first documented creature to have suffered extinction as a result of climate change. 9 Its habitat on mountain hillsides in Costa Rica was severely affected by loss of moisture that had come in the form of mists and fogs. An average temperature rise pushed cloud levels higher, to the point that these animals lacked the environmental means to survive. This story goes further than the loss of one beautiful, rare creature in nature's spectrum however.
The trick is that toads, frogs, and small lizards can often serve as harbingers for the types of problems that we ourselves might also become susceptible to. In a roundabout fashion, the golden toad guides us, like a totem spirit animal might, to a crisis that is dear to, if not particularly well understood by, many students in schools across the country - the crisis in Darfur. Of the many complexities of the human condition in sub-Saharan Africa, which is actually being slowly transmuted into the Sahara proper, is the fact that a dramatic change in weather patterns has led to the loss of annual rains across a swath of the continent. This shift has driven those who were dependent on life-giving rainwater to find resources in other regions, where, in conjunction with many other complications, conflicts have arisen between refuges and former inhabitants.
In this sense, climate change is a major contributor to the desperate situation in the Sudan. 1 0 A human crisis that students can emotionally sympathize with and intellectually come to understand as a problem that may be articulated and, hopefully, be mitigated through the use of mathematics, among other skills. I am not promoting horror as a motivator for students to better learn math, but a responsible and thoughtful examination of these kinds of interconnected environmental contexts can establish a valid urgency in their learning.
The second objective I am offering is the expectation that the use of cogent, valid environmental contexts will improve students' comprehension and application of mathematics skills, beyond the abilities that would otherwise be learned statically or in less meaningful contexts. G. Polya made the case for using the very basic context of the walls, floor, and ceiling of a classroom, which is typically cubic, or hexahedron-shaped, as an environmental context for students to better understand the geometric principles that applied to a parallelepiped of the same shape. 11 Seeing is believing, and students are better able to learn about what they believe in and trust with their own senses. This basic fact can be lost on us as adults because we have already made the leap into abstract thought so long ago. Environmental contexts are, by definition, around, and even within, us at all times. Even in the seemingly intangible air, there are opportunities to engage students, provide them some touchstone to ground their work upon, and thus improve their application of skills. This goal is valid for the improvement of students' mastery of skills in the abstract concepts that fill both algebraic and geometric studies.
My third objective is to improve students' abilities in choosing the most appropriate math skills to employ in defining, evaluating, and solving problems and problem solving. To separate the "skill to choose the right skills to use" from the previous objective of "using the right skills properly" is more than a slight nuance. It is, in fact, the division between independent mathematical competency and mere, though still profoundly important, ability to follow through algorithmic processes. Providing environmental contexts can vastly improve upon students decision making in how to go about solving problems because these contexts offer valid examples of situations that don't come with "how to" guidance. What these situations do provide are a finite set of clues, which students can begin to build interpretive skills to manage in parallel with the mathematical processing skills that form their toolkit to choose from.
An example of the important ability to recognize factors, which are not immediately or explicitly stated, can be found in another zoological note. Victoria Fabry, an oceanographer and expert in pteropods, ran into a problem during a 1985 study of how these tiny animals' form their calcium-based shells. She found that if she placed too many of the creatures in one container they would not grow shells - rather their shells would dissolve and the pteropods would die. At the time, Fabry did not fully appreciate that it was the concentration of carbon dioxide in the containers of seawater, building up from the animals' waste, which was killing them. 12 But the situation would allow her to consider the math skills she might employ and find the limit of how many creatures she could place in one container for a given amount of time without killing them. If she had pursued this modeling with inequalities she might have been able to calculate a rate of the animals' processing of oxygen into carbon dioxide. This is an example of the kind of problem solving that can help when unexpected situations arise. In other words, how to figure out… how to solve a problem.
The fourth main objective takes yet a step further by seeking to improve students' abilities in applying mathematics skills in pragmatic ways to non-scripted and, as yet, not fully understood problems. After some relative success in establishing the earlier goals of engaging students in the urgent need to build math skills, improving their abilities in applying math skills, and improving their decision making about when and how to apply skills to a given problem, there is a new frontier. This grand adventure hits on the highest cognitive levels of analyzing, synthesizing, and evaluating problems. Actually it involves nothing less than asking students to go out and look for trouble. I say this somewhat tongue-in-cheek, but in all seriousness the premise of solving unscripted problems is for students to critically observe an environmental context and find the problematic issues inherent therein. This is the ultimate objective that I am proposing - legitimate, versatile mathematical competency of students. It is very important that this high cognitive goal comes as number four of four objectives that I am outlining here.
In past school years I have asked students to simply indentify, in a few sentences, a problem that they saw in the world around them and a way in which math might be applied to solve it. Despite some minor successes, these instructional efforts have been a bit floundering, quite honestly. Students require a fluency in mathematics in order to interpret and communicate a real life situation in terms of math. Asking students to make mathematical sense of what they see around them can only be a fruitful activity if students are already performing with some proficiency in a variety of math skills that they know how to apply. As with language skills, which cannot be developed in isolation of the world which they are used to articulate, define, and communicate about - math skills must first be developed in environmental contexts if students are expected to later call upon their skills in new, untested, and novel problem solving scenarios.
Assessing Environmental Context Objectives
Measuring the achievement of these four objectives may seem an even more challenging task than introducing environmental contexts as a whole. This concern can be allayed by recognizing that each objective above is, essentially, a general enhancement of existing, more specific math objectives (e.g. students will be able to accurately draw a graph of the given linear equation in the first quadrant of a coordinate plane). For the more specific objectives there are established state standards and many existing and/or easily developed forms of assessment (e.g. provide students with a linear equation in slope-intercept form and rate their ability to accurately produce the graph). Therefore, the assessment of how well students' skills and abilities are improving, given environmental context in instruction, can be facilitated through the assessment of existing skill objectives.
The objectives I have outlined may also be directly translated to existing requirements in Pennsylvania's Math Standards. The section of standards for eleventh grade math proficiency in Problem Solving opens with a statement that is a perfect paraphrase of my objectives: "Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems." 1 3 From this broad objective, it is possible to move forward by assessing objectives from the standards for math proficiency in Algebra and Functions 14 and Geometry 15 on a more specific, per lesson basis.
If students are meeting the existing state standards for the given mathematics curriculum, after receiving instruction that is rich with environmentally valid context, then a basic correlation can be established between the context and achievement. To go deeper, it is also possible to ask students to self-assess their efficacy and ability with math skills - in relation to the environmental context that has been part of their instruction. This is a particularly good method for assessing the second objective (to improve application of skills) and third objective (to improve choice of skills), because students will, in effect, be evaluating their own application and choices. This can also serve as an additional reinforcement for student retention of those skills that are being considered. At worst, students will receive the core content of their mathematics courses, with the bonus of some environmental perspective added. At best, students will exhibit improved proficiency levels for all skills that have been adapted with some degree of valid environmental context.
For either the first objective (to engage students with urgency) or the fourth (to improve skills in novel problem solving) additional tasks or techniques may help teachers assess student success. I would particularly recommend the use of, and will personally make regular use of, formative or summative writing assignments that are sequenced into course work to create a regular feedback loop for students' affective and conceptual achievements. This is a topic that will develop in the strategies below.
Ultimately, I hope to both enrich and improve my students' learning of mathematics skills by showing them that they are the means of understanding and solving problems in the environmental in which they live. Ideally, I want students to be motivated by the sense that mathematical literacy, like the consequences of climate change, is a subject that can be neglected only to our own peril.
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