Classroom Activities
The following two activity outlines (Algebra 1 - Air Quality and, more briefly, Geometry - Green Building) are meant to be amenable to adoption by other secondary teachers of mathematics, in either Pittsburgh or in other districts, or at other grade levels with appropriate alterations to the sequence of content, levels of context, and/or levels of rigor, depending on the needs of those teachers and their students. These outlines may also serve as mere notions of how a teacher may apply other environmental contexts that are more valid for math instruction in their region.
Algebra 1 - Air Quality Project
Functionality is arguably the most important, central concept in algebra. You put one value "into a function" and you get one value out - with a unique output for each input of the function. This simple core can quickly revolve outward to the concepts of substitution, direct and inverse relationships, linearity, quadratic and exponential behavior, etc., but functionality is always at the origin point of understanding algebra. The skill sets of algebra can all be differentiated and categorized in terms of tasks and actions performed on functions: interpreting, articulating, applying, analyzing, changing, manipulating, synthesizing, creating, comparing, and, ultimately, evaluating.
Climate change is a highly complex set of concepts, but similar to algebra in that when it is sifted down to its observable, measureable parts, it is also fundamentally about functionality. For each topic within climate change, correlation and causation become the main story that environmental science has to tell us. Therefore, opportunities to demonstrate functionality from environmental context are innumerable.
Air quality is a strong example for me to choose from the topics of climate change, for the reasons of validity and regional concern that I have mentioned above. I will apply it in the form of a series of introductions and an overarching project, which are primarily aimed at the first two (of seven) units in my Algebra 1 curriculum. 19 However, to help students connect the disparate forms of math skills they will study in Algebra 1, all subsequent units of study will use some, ongoing low or middle level environmental context of air quality.
Unit 1: Patterns in Numbers and Figures
This first unit is meant to both enrich student learning of preliminary algebra skills, as well as prepare students for their second unit of study. The three keys given for this unit are: students will be able to 1) use algebraic expressions, equations and inequalities; 2) model real-world problems; and 3) apply algebraic properties. This is a ripe opportunity to begin students in practical expression of their environment, of which the air that they breathe is a primary element.
There are three steps to the basic methodology that I will use to introduce air quality context in this first unit. First, to have students articulate and express functionality and patterns that they can observe in their environment, using common language and terms of expression. Second, to have students translate their linguistic expressions into number sentences and symbolic expressions that capture the meaning of their original observations. Third, to have students practice making direct number sentences and symbolic expressions from novel observations - thus guiding students in use of a triangular representational skill set between observation, language, and algebra. This model draws from numerous pedagogical methods of associating concepts and the basic premise of language acquisition promoted by Piaget and other constructivist linguists.
Examples of student assignments will include: creating expressions for the volume of cubic air it takes to fill a classroom, school, or city block; creating equations that estimate how much carbon is emitted in order for students and faculty to arrive at school each day; creating inequalities for estimating the minimum percentage of oxygen necessary for healthy biological function; using calculations of environmental air quality to prove the importance of following the proper algebraic order of operations (improper work would lead to "life and death" consequences); and relating algebraic properties, such as the distributive property of multiplication, to distribution of air pollutants.
As the curriculum that typically introduces students to the high school mathematics continuum, the first unit of Algebra 1 also allows me the opportunity to refresh students' skills with the notation and use of fractions, decimals, and percentages. Air quality context is ideal as it provides the chance to consider the percentage composition of elemental and molecular gases in the air, differing percentages of composition and density between levels of the atmosphere, pollutants, etc. Because the air is a concoction, full of so many parts and types of gases and particles, it is also full of forms of fractional, integer representations (such as micrograms per cubic meter) to build students' sense of numeracy upon. A serious skill set for all future math study.
Unit 2: Patterns in Data
The two keys given for this unit are fundamentally related to basic statistical analysis and measurement of central tendency (i.e. averaging): 1) use single variable statistics to analyze and compare data sets; and 2) use graphical representations to analyze and compare data sets. This unit is a conceptual goldmine for considering air quality and, in turn, air quality is a powerful context from which students can improve their comprehension and practice of statistical reasoning. A specific advantage of using air quality as a context for this unit is that Pittsburgh has a wealth of locally relevant documentation. A wealth of studies in physical and social sciences, as well as public policy, are available to present true example applications of the skills that we are asking students to perform with statistical reasoning in this unit.
I will begin this unit by presenting students with several local newspaper and magazine articles that report on poor air quality issues in Pittsburgh. From these articles I will present a series of questions about the claims being made, which will include, "How do we determine what the 'best' or 'worst' air quality is?" and "How do we know what average air quality should be?" These questions will be designed to generate an objective inquisitiveness in the students' approach to interpreting statistics.
I will follow this segment of exploratory reading and discussion by having students generate a list of terms that they would need to understand in order to be able to make sense of the information on air quality. The terms "mean, median, and mode" will be ones that students have already had contact with in their middle school mathematics curricula. What will be new for most students will be the concepts of "standard deviation" and "margin-of-error." The ability to analyze data in these advanced terms will be invaluable for their future navigation of the statistics that are presented to them on a daily basis. Using air quality studies as a context for learning about standard deviation is a powerful way to reinforce student comprehension. Students will become legitimately competent at interpreting statistical information by being able to toggle back-and-forth between their own observations, which they will make of data in our activities, and their rationalizations of standard deviation, which they will be able to calculate with math. 20
Early in this unit I will have students participate in a central activity to quickly and simply produce valid, observed data sets. I will have students relax for a moment and then count their own breaths over the course of 30 seconds. The class will then report their individual counts - allowing us to produce a cumulative data set, out of thin air. Although this data set will have been obtained so easily, from so little substance, it will hold much more meaning for students because they have made it as a function of their very living and breathing. In this way air quality as a context can be an innocuous tool to significantly strengthen attainment and retention of concepts.
Once this data set is established, exercises can be carried out with it, just as they would for any set of fabricated, contrived, or arbitrary numbers (in past years they have been generated by measuring height and femur length of students, while this has been valid to true context and creating a correlation, it has been limited in expansion to more meaningful application). Students will find the mean, median, and mode - as well as discussing which of these three kinds of average might be the most valid for estimating the average rate of breathing for the class. They will identify any outliers and interpret what these distant abnormalities might signify. And, from the mean, they will develop the measure of standard deviation and margin-of-error. These are exercises and content that any students would perform in this curriculum. However, placed in the context of air quality, students will address the fact that with every few moments their bodies are taking in particulate matter from the environment around them and, again, because the students will have been intimately involved in the measurement and collection of the data set they will be better engaged and invested in the work.
As the groundwork of student awareness and vocabulary in statistical modeling is established, I will extend these advantages by making use of data from the source reports that the news articles, read earlier, were based upon. Source materials will provide mathematical enrichment to the data students have learned about. They can also benefit greatly from the very process of moving from statistics published in popular media to directly examining the objective mathematically based documents for themselves. Allowing students to draw conclusions independently from real data is an ideal, which I will continuously work toward as I refine the process of introducing environmental context to my mathematics instruction.
These reports on air quality are also robust resources of the graphical representations that we want students to be proficient working with: histograms, box-and-whisker plots, scatter plots, and functional graphing. Primary to all of these are the same measures of central tendency that make the core content of the unit. Examples of primary documents that students will use are: the State Of The Air 2009 report from the American Lung Association (2009); Chapter 5 - Inhalation, of the Environmental Protection Agency's Exposure Factors Handbook (1997); Chapter 6 - Inhalation Rates, of the Environmental Protection Agency's Child Specific Exposure Factors Handbook (2008); and the Children's Exposure to Diesel Exhaust on School Buses report, from Environment and Human Health, Inc. (2002). These resources are excellent tools to show students that their algebra skills have real ramifications in understanding their health and the living conditions in their city.
Subsequent Units
Students will continue to refer to air quality, at least at a low level of context, as they work in many aspects of organizing, solving and graphically representing linear functions, which makes up the bulk of the Algebra 1 curriculum. Too often, because the skill sets "look different" to students, they have difficulty maintaining a link between the patterns in data that they begin the Algebra 1 course with and the extrapolation of functional significance that they end the course with. The resonance of air quality throughout the course, with frequent reference to prior knowledge from the first two units of study, will facilitate continuity and conceptual depth for students of all ability levels.
The most significant achievement of using air quality as context for Algebra 1 would be to have students autonomously conduct a facilitated study of the air quality. Such a study could borrow from the methodology of the Children's Exposure to Diesel Exhaust on School Buses report from Environment and Human Health, Inc. (2002) or take a far more modest approach to data collection and development of exposure findings.
Depending upon time and resources, this project could be conducted to measure rates of particulate matter in our school building, on the public city buses that the majority of our high school students use to commute to and from school, on the school buses that our middle school students use to commute daily, and on the downtown sidewalks where our students inevitably spend a significant part of their days - exposed to whatever the city's air holds for them. This project could begin with unit two of the course and be incrementally developed, so that the data could be applied to the skill sets that make up the content of the third through seventh units.
Geometry - Green Building
Geometry is built, for all practicality, on the fundamentals of building. The development of human civilization, in its most basic structure, has relied on at least rudimentary geometric reasoning since the beginning of ancient history. Now, on the eve of revolutions in the construction industry impacted by growing recognition of economic and climate change issues, geometry is no less essential.
What I modestly and briefly propose is to make consistent low and middle level context of LEED standards 21 and green building concepts for students in the first four units of our Geometry curriculum, and one, summative, high level context project in which students will assess our own school according to the documents for new construction and major renovations. This will consist of me reducing the complex certification document checklists down to basic geometrical principles that are applied within them. In this way I can present students with palatable notions of true application of what they are studying. It is my intention to include talks given by local designers and architects who are versed in green building codes, to take students to at least one LEED certified structure in the vicinity of our school, and to have students apply their geometry skills to a proposal for optimizing student and faculty usage of our school building.
This series of contexts can be applied to the core of geometry study (e.g. basic elements such as points and line segments, measure and properties of angles, midpoint and distance formulas, etc.) and can be expanded upon through the study of geometric proofs and logical argument. There are also many aspects of civil management of our downtown environment, such as the zoning of commercial property in proportion to public space, which might be explored via the linguistic tools of conditional statements such as the reflexive, associative, and transitive properties.
I will go on to reference some real health and social issues of our city. Ideally this will include some interaction between students and city representatives. Because ancient Mediterranean civilizations are so reputed in the study of geometry, I think students will also better understand how mathematics impacts their everyday lives by considering their connections to our city. This is most fitting for my students, who are focused in the arts.
Athens was a place where the life of the individual and the life of the city were not split into separate places. Theater, sculpture, music, poetry, business, ath letics and philosophy were part of the city's life and part of the individual's life. A great temple like the Parthenon belonged to each citizen and to the city as a whole at the same time. 22
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