Running Head: Student Definitions of Mathematics

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Student Definitions of Mathematics

Running Head: Student Definitions of Mathematics

Student Definitions of Mathematics:

A Co-operative Inquiry Study in a

Developmental Mathematics Classroom

By

German A. Moreno

NMSU-Dona Ana Community College

Chapter 1

Introduction

Mathematics educators have been challenged over the last several decades to find ways of improving outcomes for their students for many reasons-most of which involve external pressures and the anxious concerns of agencies and institutions. The resulting initiatives have pushed educators into roles requiring them to promote curricula which in the end weaken the very society they wish to strengthen. In particular, students of ethnic minority descent, struggle to fit in into this mainstream paradigm. It is time now for mathematics educators to help all students reclaim mathematics so that the curriculum better captures the essence of what mathematics really is. This paper will look at ways in which students and their teacher use a blog to co-operatively examine notions of what mathematics really is. As this research is done in a predominantly Mexican American community, we also aim at examining ways in which culturally relevant perspectives can be integrated into content with the intent of freeing students from authoritarian views of mathematical knowledge.

Researcher as Instrument Statement

In transitioning from teaching high school mathematics to developmental mathematics at a community college it has become apparent to me that there are certain errors made by contemporary applications of learning theories and personal ideologies. As somebody who formerly embraced some of these ideologies (such as constructivism) I now believe that though well intentioned there were missing components to my application of these ideologies.

As a Mexican American who went to public school in El Paso, Texas, I vowed that my students’ experiences would be different from mine. The high school that I attended was essentially a homogenous Mexican American community. The preferred teaching method used by my teachers (who were mostly Anglo) was the direct teaching method, and it was my assessment that this expressed our teachers’ lack of confidence in my and my classmates’ ability to develop intellectually. Unfortunately, I remember very few teachers at my high school, who expressed an alternate philosophy.

Demoralizing speech acts such as critiquing my ability to incorporate new language still linger in my mind. Memories of a mathematics teacher who walked out of class during exams so as to allow students to “cheat” still perturb me as they too signaled to us our teacher’s lack of belief in our ability to learn. In response, I eagerly searched for a different way to teach my students.

As a result of interactions during professional development and through my training in graduate courses geared toward mathematics educators, I found something which I thought had meaning for me as a teacher working in a diverse environment. This is where I first encountered constructivist methodologies which I quickly adopted.

Several years have passed since I taught at the secondary level and looking back has allowed me to be objectively critical of my own work as a high school teacher. Constructing spaces that allowed students to discover and play was did in fact seem like a successful scheme, as it appeared obvious to me that particular students flourished under such environments. As my interest and interaction grew with certain students their development and growth seemed to validate the theoretical paradigm.

Yet there were faces in the background that I now realize were not being served-faces whose realities never completely surfaced during their term in my classroom and who were ultimately left to fend for themselves in the mire of the mathematics classroom.

The results were not realities for me until I stood before the students in my developmental mathematics courses at the community college. Given the direction that mathematics teaching has taken in the last twenty-years, I should have expected better prepared college student. But the reality is harsh. Over 90% of our incoming freshman at Dona Ana Community College must begin their mathematics coursework with what public schools consider to be 8th grade mathematics. The students, whose emancipation was promised by constructivist methodologies, had not gone away and transformed themselves as the paradigm seemed for me to promise. My old students whom I had ignored before seemed to be returning to me to respond satisfactorily this time.

Chapter 2

Literature Review

History of Mathematics and the Modern American Student

We begin our discussion on how to address these students with a very brief look at the history of mathematics. The connection between mathematical thinking and human perception and experience has been present as far back as accounts of mathematics have been recorded. As early as 20,000 B.C., man has been counting, as is evidenced by artifacts discovered in Zaire (a count that is thought to have been of certain periods of the moon) (Katz, 1998). Man has always wanted to quantify his experiences. One need look no further than yearly agricultural cycles in seasonal changes or monthly cycles connected with fertility to find examples of things which men wished to quantify. Our calendrical methods for numerically keeping track of time can be found in many original sources and cosmological cycles and their relationship between the aforementioned have been of great interest to humans throughout the ages (Katz, 1998; D’Ambrosio, 1999).

As can also be found in historical sources man has also used mathematical language to refer not only to cosmological entities, but to quantify things directly related to the daily human interactions of men. Babylonian papyri sources going back 3000 years describe men for example, dividing up loaves of bread and inquiring (and correctly concluding) relationships between areas and perimeters of land, and the volume of pyramids (Katz, 1998; Midonick, 1965).

Not only has mathematics been of interest because of our ability to use it as a way to give descriptive language to these external entities and experiences, but the language itself has been of interest to man because of its direct link with something which is exclusively human-i.e. the ability to reason. Although much of the evidence described in the previous paragraphs shows that mathematics was used to describe experienced events, many of the artifacts found display examples that are de-contextualized or whose context is contrived (Katz, 1998; Midonick, 1965)-an indication of mans belief in the intrinsic worth of mathematical thought .

Evidence that the peoples of the Americas mathematized their world too exists (Ortiz-Franco, 1996). The organized life that we call civilization in the region began approximately about 5000 years ago. The earliest evidence of numerical inscriptions using the positional systems of bars and dots has been traced to the Olmecs in approximately 1200 B. C. (the historical significance of this date being that it preceded by some 800 years accomplishments of Aristotle, Plato, and Euclid, whose western culture did not yet have a positional number system). The Mayas developed their amazingly complex calendar system and astronomical sciences around this mathematical system hundreds of years before Galileo and Copernicus lived. We know that for 1600 years before Columbus accidentally arrived in the New World, the Mayas wrote and had kept thousands of books in which they recorded their history and cultural achievements. We also know that Mayan astronomers calculated the cycles of the heavens so precisely that they could predict solar and lunar eclipses to the day hundreds of years before they happened (such as the prediction of the solar eclipse that occurred on July 11, 1991).

These early accounts show that everywhere on earth (including in the Americas), man has always been concerned with mathematizing his world. Unfortunately, this is something that is almost entirely obscured by what is done in mathematics classrooms today. The last 100 years of education have done a spectacular job in presenting mathematics as an object of learning whose worth is only related to our ability to use it as a tool which serves the human technological imagination. While some convincingly argue for the necessity of doing this due to the challenges and threats from abroad-such as the launching of Sputnik in 1958 and the current challenges of globalization (Friedman, 2007; Garret, 2008)-we seek to ultimately examine curricula which present alternative views of mathematics. Thus, we propose that it is time for our focus to change-especially in communities where mainstream curricula is failing- toward one that is not espoused by our institutions but one that truly serves the needs expressed by and arising out of the concerns of our students.

Current Mathematics

Burt asserts that modern curricula are primarily dominated by two “subcultures” one individualistic and another which is sociologically focused (1990). Burt believes that behaviorism, Piagetian theories, cognitive psychology, focus too much on the individual (Gordon, 1990). On the other hand he also affirms that although sociologically based movements have made some strides in de-emphasizing the notion of the individual as the instructional unit, this paradigm does not go far enough because it is not critical of the culture that produced the content in the curriculum (or of the cultures learning the curriculum). This paper wishes to examine these “sub-cultures”/movements which result in the application of what I believe are two oppositional philosophies (direct teaching and constructivism), which often produce a mathematics curriculum that either reduces student autonomy or which ultimately separates weaker students from those whose backgrounds give them advantage.

Teachers, adopting a direct teaching philosophy perpetuate this division in several ways. First, they do so by emphasizing a propaedeutic curriculum that rarely allows students to do actual mathematics. They emphasize algorithms over problem solving. While developing automaticity with algorithmic processes is important for students who will eventually solve higher-order problems, manipulating marks on paper cannot pass as mathematical acts (Papert, 1998). Secondly, teachers who prefer this paradigm teach problem solving as a step-by-step activity. When educators teach problem solving as a procedural activity in which only standard problems with standard techniques are learned, educators deprive students of the ability to develop autonomy in the mathematics classroom.

On the other hand, teachers whose preferred paradigm is constructivism, err too. Aside from the philosophical dilemmas regarding absolute relativism (Tilich, 1963)-a component of the constructivist paradigm-there too exists much empirical evidence that contradicts constructivism as a learning modality (Kirschner, 2006). More importantly however, that is as it pertains to this paper, there are broader curricular problems that are often ignored by constructivists. By focusing for example, on contrived “real world” problems and on numeracy (results of an experientially focused curricula), teachers highlight the notion that the only mathematics that students can and should learn is that which is experienced on a daily basis-i.e. utilitarian mathematics (Klein, 2007; D’Ambrosio, 1999). By not allowing students to participate in mathematics whose outcomes extend beyond utilitarian purposes, students are denied participation in their historical and human legacy. A society which is committed to democratic values cannot accept this outcome (D’Ambrosio, 1999)-an outcome which I believe is an unfortunate consequence of many constructivist programs.

A New Direction in the Digital Age

Due to the nature of our changing world it is important to reconsider the mathematics we teach and how we teach it. Mathematics education can no longer be only thought of as an individualistic task or even a socially constructed body of knowledge which focuses entirely on content-especially when our inter-dependence continues to increase worldwide as it has done in the last 20 years due to the advances in digital technology (Freeman, 2007). It is time now to consider the possibility that mathematics educators can make an impact that transcends beyond the limits of an individuals’ minds. This section will consider how technologies mostly ignored by mathematics educators can be used to achieve this goal.

Although technology has several roles in structuring activities and spaces which must be recognized by teachers, of primary importance regarding the focus of this paper is the technology that acts to structure local and interpersonal spaces. Connectivity tools, in particular, allow us to interact with each other with an ease never seen before. Whether we are trying to communicate with members of our local community or whether we are connecting with individuals or groups of individuals across the globe, communication technology is structuring and formatting our everyday interpersonal activities and spaces (Friedman, 2007).

Mathematics teachers should not ignore these technologies even if they seem unlikely vehicles for learning content. While teachers of mathematics have traditionally been directed to use internet sites which use interactive graphics such as JAVA applets, that type of technology focuses on individual cognitive goals rather than on the cultural value of mathematics that this paper seeks to explore. Before learning the content teachers should be aware of their and their students’ intentions for learning such content. This is something that could easily be explored using the interconnectivity tools of the internet.

Blogging is one communication tool that teachers can easily set up and which can be used to begin discussions (Poling, 2006; Ray, 2006 ). The classroom model of blogging, when used appropriately, can help build communication and collaboration among students. By blogging, students can learn from each other as they make connection, ask questions, and draw conclusions while blogging. Hyperlinks, pictures and audio segments can be effortlessly added to enrich blogs. Students can be asked to post their comments to a blog, explaining what they learned about after visiting linked sites.

Students can direct their own learning as the computer brings them in contact with information (and people) not available in print (Poling, 2006; Ray, 2006). As students communicate using tools like blogs, they question and challenge each other's thinking, leading to deeper and more meaningful interactions than previously afforded during activities like individual journaling. By using tools such as these teachers can begin to re-structure and re-format their educational spaces. These restructuring changes can alter the dynamics within the classroom and can transform an individualistic paradigm to one that is more responsive to the local communities and students.

Implications and Problems

This paper argues that mathematics is presented in schools in ways that are detrimental to the cognitive and personal development of students. We believe that cultural relevance and broader educational goals are not currently taken into consideration. Furthermore, a focus on utilitarian mathematics in public schools creates learning hierarchies that reserve higher order mathematics for privileged students. Finally, because the curriculum focuses on individual performance via unproven methodologies and/or those supported by weak philosophical arguments, students of ethnic minority decent suffer. The data collected in this study is meant to determine whether students’ definitions of mathematics support or contradict these propositions, and whether other avenues of discussion are relevant to these issues.