CEP 800 Podcast Assignment

I was not quite sure what to expect from the students that I chose for my podcast. I wanted to understand how students perceive learning mathematics and how their conceptual understanding of the learning process aligned with the material that I h ave been reading for my graduate course. I did not want to lead my students in their thinking;therefore, I did not provide them with questions that guided them in one particular direction. Each interview lasted approximately ten minutes, but I realized that I should have had more recorded during the editing phase. I am passionate about film-making and I should have realized that I needed more coverage.

I found it quite interesting that the younger students focused on mathematics as skill development. They felt that although students were "wired" differently at birth, they could match advantageous genetic make-up by working harder than those students. I believe this line of thinking works well with the behaviorist model by using reinforcement to produce desired effects; the result is the learning of mathematical concepts. The Pre-Calculus student has had more experience with mathematics and felt that math is conceptually complex. He discussed the idea that mathematics is quite abstract and that genetics dictated which types of mathematical concepts would be able to be learned by the individual. Our course readings have discussed the behaviorist model and its inability to account for abstract thinking such as Einstein's. This student's comments regarding the abstract nature of mathematics would accompany the detractors of the behaviorist model of learning.

I figure that the students would understand how they learned, but I was impressed that they had formed arguments for how other students learn as well. I now wonder if there is a correlation that exists between how these students feel people learn and how people succeed in life. The younger students feel that hard work can offset genetic deficiencies, and my Pre-Calculus student initially stated that genetics play an overwhelming factor in the developmental process of a mathematics student. Do they have similar feelings about social class structure and societal/economic mobility? If so, what has led them to have these feelings?

Blog Posting Assignment 2

Technology transcends itself from novelty to an educational tool only when it addresses problems of practice. I have learned that you have to look at the educational need(s) that you are attempting to address and then decide which pedagogical strategies and technology can best serve your needs. The TechQuest project helped me develop skills related to the analysis of technology for educational purposes. It is important to understand what a piece of technology can do for you before you add it to your teaching arsenal. The SIG and TechQuest project relied on thorough educational research and the use of Google Scholar made it much easier to identify quality journal articles related to the different research topics. The search engine has evolved since I first used one about fifteen years ago, and I am curious to see the next iteration in Web 3.0.
I have become accustomed to using Google Documents for most of my office needs due to the ability to access the files from any computer connected to the internet. In addition, I have integrated the use of this software into my classroom due to its ease of use for collaboration. Collaborative creation is a direct result from the development of Web 2.0 technologies. The constructivist approach is a more accessible pedagogical strategy than it ever had been before.
The goals that I set for myself in regards to learning about technology integration are unreachable in theory. My goal to become more knowledgeable in the use of technology to address problems of practice is a never ending pursuit; yet, I have made considerable progress over the past three graduate courses. I have learned a great deal from the different group members that I have worked with as well. It has been great to observe the viewpoints that are coming from different professionals that are specialized in different content areas than mathematics. I have missed being in an educational setting with peers, and look forward to continuing the MAET program. I believe that an effective educator must always be evolving in order to keep up with the educational trends. I am a life-long learner that revels in furthering my own education and then applying the knowledge in my own teaching.

The concepts of the two large assignments (TechQuest and SIG) will be an integral part of my future development as a learner and teacher. I will continue to seek out technologies that can help address my problems of practice and become part of the global conversation with other educators. The internet has provided the platform for professional collaboration, and I hope to continue to join others' in the quest to develop educational resources for use by me and other educators. The knowledge that these communities can provide is exponential. I could be nearsighted and attempt to develop educational objects myself without the input of others or I can let others tinker with these objects and enhance them with original thought. I will continue to modify my goals in an attempt to become more of a global collaborator and continue to develop my educational technology analysis skills.

Project Evaluation

The TechQuest will continue to evolve as I address problems of practice with the unique types of functions throughout my career as a mathematics teacher. I will continue to explore the possibilities of GeoGebra as an effective tool for representing mathematical functions and continue to read current related research. I have become comfortable using the process laid out during this TechQuest will apply the algorithm to additional TechQuests in the future. My new familiarity with scholarly research engines will make further research time efficient and focused. I have learned and sharpened the skills necessary to analyzing technology for its pedagogical use. The abundance of technology that exists on the web is staggering and can prove to be overwhelming. The structure of the TechQuest helped develop the skills necessary to applying technology to educational concerns. Student learning should always focus the technology, and it seems that is not often the case. The TechQuest began with the identification of the problem of practice and followed with finding the appropriate technology to address the problem. We, as educators, often want technology that can do a million different things, but we should be focusing on technology that addresses an educational need in a way that cannot be done as effectively otherwise. Technology should not be brought into the classroom just for the sake of having technology.

The next time I proceed in a TechQuest project, I will pick a problem of practice that is more specific. The concept of the function is very broad and covers multiple years of mathematics. I should have focused on one particular type of function such as the linear function or sinusoidal functions. I believe that narrowing the project would result in a project that has a smaller development cycle. Educators continue to reflect upon lessons and technology and shape the malleable lessons we create. This TechQuest that I have begun to implement will continue until I retire as a teacher. Technology will evolve and problems of practice will be addressed in a plethora of ways, and it is my job as an educator to continue to improve upon my pedagogical strategies.

It is estimated that the modern human originated 200,000 years ago, and it is believed that written word started around 5000 years ago. Humans did not have the necessary transference of knowledge to develop the written word systems for 195,000 or so years, yet, since the development of word systems we have advanced technology to the point that we can instantly share information across the globe. The advent of the internet has allowed the sharing of information among students and professionals alike, and we now have the ability to evolve education by creating a network of highly informed educators by sharing content. The internet is providing a global landscape in which educators can collaboratively research and create educational content that is based on research-proven educational strategies. I would like to become part of the larger discussion and help create and share content that other educators can use as well.

TechQuest Implementation Blog

TechQuest Annotated Resources

I decided to make GeoGebra the dynamic software for my TechQuest. You can read my prior two posts to gain some insight into the design of my TechQuest. I will provide a description of my research method along with an annotated bibliography containing five the of the resources that are shaping additional research and the further implementation of my project.

Research method: After identifying GeoGebra as the tool i would use to address my educational concern, I refined my research by using a simple Google search to see the common results related to using GeoGebra to help aid in my quest to have students gain a deeper understanding of the mathematical concept of the function. This attempt of a basic search did not yield anything that I would later cite; therefore, I moved on to using Google Scholar to focus my search on scholarly articles and periodicals. I am comfortable using Google Scholar and find easier to use than some of the university engines used. GeoGebra is a relatively new product and is gaining momentum as an educational tool; therefore, research is slowly amassing to analyze its effectiveness. I limited my parameters to include only articles related to mathematical functions. A funny thing happened during my search for sources; I found that several of the articles were linked to a publication index that GeoGebra has on the wiki-page that they host on the organization's website. I found many great articles that related to the nature of my individual research (the function). I found only a handful of instances where the pedagogical shortcomings of GeoGebra were discussed using Google Scholar. I will continue to research for GeoGebra centric lesson plans, because I did not find any detail lesson plans related to what I am attempting to do. I did find examples of other technologies used to tackle similar situations, but I have not included them below.

Surprises: The software is used internationally and research is being done across the globe. I did encounter several articles that seemed to have grammatical errors in their titles, but the content was appropriate and did not contain these types of errors throughout. I found it interesting that some of these articles were published in periodicals. For example, I found the article Applications GeoGebra into Teaching Some Topics of Mathematics at the College Level by Diković. The article is from Serbia and I am sure that the translation is responsible for the awkward title.

I believe that my strategies were successful, and was aided by the My annotated bibliography follows below.

TechQuest Annotated Resources:

Hohenwarter, M. (2006). Dynamic investigation of functions using GeoGebra. Proceedings of Dresden International Symposium on Technology and its Integration into Mathematics Education 2006. Dresden, Germany: DES-TIME. Retrieved from http://www.geogebra.org/publications/2006-DES-TIME.pdf.

From the abstract: "This article describes the mapping and covariation aspects of functions and shows the dynamic ways of investigating functions with the free educational mathematics software GeoGebra." The abstract provided in this article is a near summation of my TechQuest. The content of Hohenwarter's article contains explicit use of GeoGebra's software to explore the mathematical function. The exploration of the different function types is germane to the educational concern I am addressing with my project. Hohenwarter's name appears on several resources related to the use of GeoGebra as an educational tool. GeoGebra offers advantages to students than traditional methods. Hohenwarter is the creator of the software and he understands the educational concern and how his software can help alleviate the problem. The content of this article will help shape my project due to its understanding of the concern and examples of how to use the software effectively.

Sangwin, C. (2008). Geometrical functions: tools in GeoGebra. In: MSOR Connections Vol 8, No. 4, 17-20. Retrieved from http://ltsn.mathstore.ac.uk/headocs/84Sangwin_C.pdf.

This article discusses GeoGebra's dynamic approach to the mathematical function. The free software allows both the Algebraic and Geometric representations to be displayed simultaneously. Mathematics is said to contain two main components: algebraic representation and graphical representation. The concept of the function is taught throughout a student's mathematical career and involves teaching and reteaching of the concept of mathematics with these two representations. Students often struggle with the relationship between the graphical and algebraic representation of the mathematical function. This article describes an experience with teaching the concept of function using GeoGebra. This article helped me gain insight into the functionality of GeoGebra as it relates to my TechQuest.

Karadag, Z., McDougall, D. (2009). Visual Explorative Approaches to Learning Mathematics. Retrieved from http://www.pmena.org/2009/proceedings/workinggroup90649replacement.pdf.

This article discusses the agenda from a focus group focused on mathematical representation and its shift from static to dynamic with the advent of technology such as GeoGebra. The article discusses the creation of dynamic worksheets in relation to GeoGebra. Students will explore an object created using GeoGebra and have to answer questions related to the mathematical object. An object in GeoGebra is akin to a manipulative that can be posted on a website or in a file off-line. The article contains an example of the unit circle and the graphical representations of the trigonometric functions. The result provided an interactive applet that addresses the connection between the two concepts in a way that pen and paper simply cannot do.

Böhm, J. (2008). Linking Geometry, Algebra, and Calculus with GeoGebra. TIME 2008 conference, South Africa. Retrieved from http://www.geogebra.org/publications/2008-TIME-Boehm.pdf.

This article introduces the features of GeoGebra and discusses the software's use to link the content contained in Geometry, Algebra, and Calculus. This document provides real examples of the educational tool used to address issues within mathematics. The idea of a function is universal throughout each of the sects of mathematics and the presentation contained highlights many different function types being addressed with GeoGebra. Furthermore, the article discusses the probable evolution of this software of the next few years. Functions are represented using algebra, tables, graphs, and mapping diagrams. An object created in GeoGebra can get very complicated if you are attempting to introduce many of these representations at the same time. Spreadsheet software (tables) will become integrated with the tools currently available to create software that is even more dynamic.

Diković, L. (2009) Applications GeoGebra into Teaching Some Topics of Mathematics at the College Level. In: ComSIS Vol. 6, No. 2, 191-203. Retrieved from http://www.doiserbia.nb.rs/img/doi/1820-0214/2009/1820-02140902191D.pdf.

From the abstract: "This paper presents new trends in technology and learning through Geogebra, which could be especially important for the future development of e-learning for College mathematics. Also, contribution of this paper is presentation of methodological frames on several specifics examples for teaching mathematics at the college level on interactive and creative way. Discussed in this paper are the findings of a didactic experiment where some number of undergraduate students were exposed to an innovative (to the departments) method of teaching and learning a part of calculus supported by GeoGebra. Statistical analysis confirmed the fact that the use of the applets created with the help of GeoGebra and used in differential calculus teaching had a positive effect on the understanding and knowledge of the students."

This article was aimed at college level mathematics, but the content of math courses in college and high school are very similar until you get past Calculus. The article lists the strengths of the software for representing mathematics using the interactive software and compares it to other methods of visual mathematics. The article does discuss the shortcomings of the software, but are relatively obvious in nature. Research is limited on the software at this point, and it does lack some functionality that will eventually come in time. Pedagogical strategies need to be paired with technology in order for successful student learning. This article addresses the pairing of cooperative learning with GeoGebra and the benefits that would result. Diković provides examples of the use of educational objects created in GeoGebra that tackle concepts in algebra and calculus. Furthermore, he describes a study done in a college calculus classroom that resulted in positive growth for students that were exposed the use of GeoGebra. This article provided me with insight into the effective use of objects created and actual research that was done to validate its use as an educational tool. The implementation of the TechQuest is dependent on research such as this article. The development of educational tools using GeoGebra is time consuming and can be quite involved, but the possibility for improved learning opportunities makes this software a viable tool.

TechQuest Project Description

Identification of the Problem of Practice:

I am in my third year of teaching secondary Mathematics here in Michigan. This year’s junior class is required to successfully complete four years of high school Mathematics. They are expected to pass Algebra I, Geometry, Algebra II, and an additional mathematics credit. I am sure you have heard the old house analogy when someone was speaking of the scaffolding present in the curriculum. Students do need to build on prior knowledge to climb through the content. The core content of algebra requires students to represent functions in a myriad of ways. Students begin with linear functions and advance to quadratics and beyond. The concept of a function is integral to the mastery of knowledge in secondary and post-secondary Mathematics.

In chapter six of their math standards, the National Council of Teachers of Mathematics (NCTM) states “Functions are one of the most important mathematical tools for helping students make sense of the world around them, as well as preparing them for further study in mathematics (Yerushalmy and Shternberg 2001). Functions appear in most branches of mathematics and provide a consistent way of making connections between and among topics. Students’ continuing development of the concept of function must be rooted in reasoning, and likewise functions are an important tool for reasoning. Thus, developing procedural fluency in using functions is a significant goal of high school mathematics. (41)”

Students begin to question how the content covered relates to their own lives as they progress through these courses. The content requires students to master skills and knowledge that the students feel have little or no relevance to their own lives. I feel that the compelling problem of practice that I want to address in this TechQuest is: How can I help students identify the function that is best suited for modeling a given real-world situation? This is Algebra I content expectation A2.4.1, but is a core content piece throughout Algebra II, Trigonometry, and Pre-Calculus Courses. Students learn in various ways and I am searching for some technology that will help me incorporate many different pedagogical strategies. In their book Technology and to Support Learning, Bransford and Cocking state “Technology can help create an active environment in which students not only solve their own problems, but also find their own problems (Bransford 195).I am hoping to find technology that helps me create an active situation where real student learning can take place. In my own experience, I have developed related lessons that students can master the content temporarily, but cannot reproduce the content later down the line. For example, the student may be able to complete similar tasks during an Algebra II course, but will experience difficulty with a related activity once they have entered Pre-Calculus and are taking a pre-assessment. I want my students to have a memorable experience that will generate real learning related to this subject matter.

Additional Research and Resources:
In addition to the pedagogical research that I have listed above, I am using the following resources to help aid in my TechQuest: (this is not a final list of all the research that I am using to develop my TechQuest, but is what I am currently exploring)

Online Function Related tools:
http://www.geogebra.org/
http://www.ct4me.net/math_manipulatives.htm
http://education.ti.com/educationportal/sites/US/sectionHome/tutorials.html
http://www.fsmq.org/resources/using-algebra-functions-and-graphs,83,MO.html

Pedagogical research related to using technology with functions:
http://mathforum.org/~pcmi/technology11.25.09.pdf
http://www.thefreelibrary.com/Assessing+Students

Established Lesson Plans: http://www.ncwiseowl.org/kscope/techknowpark/freefall/resources.html

Plan for Implementation:
During this course: The following steps will be taken during this course:
Research (Technology and Pedagogical): I am currently researching various technologies to employ to help solve this problem of practice. I believe that it will not be just one piece of technology that will lead me to the development of an effective pedagogical tool. I know that I will use some form of database/spreadsheet software and am leaning toward using Exel, but Google Document’s version could potentially win out in the end. In addition, I am investigating the use of GeoGebra as a graphing tool as well. I have used many of the applets before, but I will devote time to learning how to program for this software as well. I believe that I should use readily available technology that is at no additional cost to the district or me. I am hoping to develop a creation component for the students as well. I want to explore the concept of students using Google Form to create assessments for their peers that gets to the heart of function model identification based on real-life scenarios described.

Development: In the next few weeks, I hope to begin the development of the lesson plan that incorporates the above technologies. I will continue to tinker with the lesson plan and develop the technology in tandem with research –proven pedagogical strategies to create a memorable experience for the students.

After this course: I hope to address the following issues:
Implementation using a Focus Group: I want to implement the lessons within a focus group of mathematics teachers and a focus group of high school students. I will give the focus groups assessments at the end of the lesson to collect for analysis.
Reflection: I will collect data and analyze the assessments from the implementation. I will reflect upon the lessons and the use of technology attempting to address this particular educational need.

Address Issues: I will address any issues with my TechQuest lessons and make changes where they need to be made.

Re-implement the lessons: I will attempt to use these lessons again after I have made the necessary changes. I have always viewed education as necessarily malleable, and that we have to adjust the lesson continuously to serve our students.

The four common places of education:

Teacher: This TechQuest addresses the concept of the teacher in a plethora of ways. I, the teacher, am responsible for the creation of the lessons that incorporate several different pedagogical strategies with current technology. The teacher is also responsible for the implementation of the technology as well as subsequent reflection.

Learner: Student learning is at the focus of any implementation of technology into the classroom. It is said that technology becomes a tool when it helps address an educational concern or problem of practice. The TechQuest will be developed with the learner in mind. It will attempt to create a memorable experience in which the students will create an artifact to showcase their learning as well.

Subject Matter: The TechQuest helps to address the overarching concept of functions within the realm of high school mathematics.

Setting: The TechQuest will be implemented in secondary Mathematics classrooms and will range from Algebra I to PreCalculus.

TechQuest Draft 1: Establishing a Problem of Practice

I am in my third year of teaching secondary Mathematics here in Michigan. This year’s junior class is required to successfully complete four years of high school Mathematics. They are expected to pass Algebra I, Geometry, Algebra II, and an additional mathematics credit. I am sure you have heard the old house analogy when someone was speaking of the scaffolding present in the curriculum. Students do need to build on prior knowledge to climb through the content. The core content of algebra requires students to represent functions in a myriad of ways. Students begin with linear functions and advance to quadratics and beyond. The concept of a function is integral to the mastery of knowledge in secondary and post-secondary Mathematics.

In chapter six of their math standards, the National Council of Teachers of Mathematics (NCTM) states “Functions are one of the most important mathematical tools for helping students make sense of the world around them, as well as preparing them for further study in mathematics (Yerushalmy and Shternberg 2001). Functions appear in most branches of mathematics and provide a consistent way of making connections between and among topics. Students’ continuing development of the concept of function must be rooted in reasoning, and likewise functions are an important tool for reasoning. Thus, developing procedural fluency in using functions is a significant goal of high school mathematics. (41)”

Students begin to question how the content covered relates to their own lives as they progress through these courses. The content requires students to master skills and knowledge that the students feel have little or no relevance to their own lives. I feel that the compelling problem of practice that I want to address in this TechQuest is: How can I help students identify the function that is best suited for modeling a given real-world situation? This is Algebra I content expectation A2.4.1, but is a core content piece throughout Algebra II, Trigonometry, and Pre-Calculus Courses. Students learn in various ways and I am searching for some technology that will help me incorporate many different pedagogical strategies. In their book Technology and to Support Learning, Bransford and Cocking state “Technology can help create an active environment in which students not only solve their own problems, but also find their own problems (Bransford 195). I am hoping to find technology that helps me create an active situation where real student learning can take place. In my own experience, I have developed related lessons that students can master the content temporarily, but cannot reproduce the content later down the line. For example, the student may be able to complete similar tasks during an Algebra II course, but will experience difficulty with a related activity once they have entered Pre-Calculus and are taking a pre-assessment. I want my students to have a memorable experience that will generate real learning related to this subject matter.