I do not believe that I could have selected a better book to help me resolve some of the questions I have had about adult education. I wish I had the time and space to talk about each essay in this collection and share my reflections on how they apply to my profession as an educator in mathematics, but I do not. Therefore, I have focused my discussion on pedagogy and andragogy as a theory, in relation to the instructor, and in content.

Pedagogy - Assumes the learner knows nothing about the subject and must be taught.

Andragogy - Assumes the learner has, from personal and professional experience,an intuitive grasp of the subject.

Throughout most of our childhood education it has been believed that upon entering a classroom that we were clean slates, with no basic knowledge of the subject. Within this assumption it was believed that the educator would have to provide each and every bit of information that was to be learned in the curriculum. This pedagogical approach would stifle the natural abilities of the student. But an andragogical approach would highlight and utilize these inborn qualities. In Mathematics these inborn qualities are intuition and reasoning, which I believe every learner possess at differing levels. It is these inherent traits that assist a student in predicting and understanding a reasonable answer before any numbers have been computed. A person's rationale will aid them in learning new mathematics in the absence of direct instruction.

Pedagogy - There is no clear need to learn the subject other than to be 'educated'.

Andragogy - The need to learn a subject is directly related to his or her life.

How often have you sat in a class and asked either yourself or the instructor how the course information would be used in your life? Is there any immediate or future value? In a pedagogical classroom this is a frequent dilemma. Disconnected pieces of information are presented with no existing real-life frameworks to connect them to. In Mathematics it is imperative to tie new concepts to previous or anticipated experiences. For example, a parabola may have very little meaning by itself, but coupled with the motion of a person jumping from a diving board or the trajectory of a baseball the parabola comes to life. Taking this andragogical approach, and using real-life situations to explain mathematics, aids the student's comprehension and retention of the concepts.

Pedagogy - Teaches a subject as a set of rules and as levels of subskills.

Andragogy - Teaches a subject for an understanding of content and concepts.

Many of us have taken a writing course where we were taught the 'rules', sentence structure, paragraph construction, and then thematical development. These classes were technical classes on how to be a 'good' writer based on a set of rules. But isn't clarity of writing and effective communication the lasting and important aspect of writing. Similarly, problem solving and critical-thinking are the lifelong outcomes of mathematics, not the rules. I do not believe it is important for a student to use a particular formula or process to arrive at an answer. If a student can utilize another method, such as guess-and-check, to obtain the answer then it is clear that they were able to critically-think about the problem and its outcome. In my courses I include a writing component in most problems so that students use multiple forms of explaining rational thought rather than just the black and white of numbers. It is the ability to problem solve that is lifelong learning, rather than a set of formulas and rules that they will forget in six months.

It is believed, although there is little supportive research, people of all ages learn better from andragogy (Fowler, pg. 12). I agree with this notion, but I can not imagine a successful math classroom in which andragogy is the sole method of teaching. I firmly believe that a blend of both pedagogy and andragogy are needed to teach Mathematics, especially in upper-level classes. There are skills that need to be taught pedagogically for successful andragogical exploration of concepts. Solely relying on pedagogy or andragogy would surely assure failure. Although both philosophies of teaching are utilized in a mathematics classroom, the hope is the student will complete the course with the lifelong learnings of andragogy. Examining learning by the rules, formulas, and direct processes a learner can recite is shortchanging the real meaning of education. Instead, education, especially from an andragogical view, is the ongoing, lasting behavioral changes: confidence, transfer of learning, growth in skills, concepts, problem solving, and critical-thinking. If a student can leave a course with an increase in these areas then the learning experience has been successful. I can not remember the exact skills I learned in many of the undergraduate classes I have taken, but I can count the long-term behavioral changes I have experienced. What is interesting about this is that many of the classes I took were taught pedagogically. I believe the reason I gained these behavioral changes is that I was a self-directed adult learner. "Andragogy assumes that adults are self-directed, capable of diagnosing their own learning needsThey are even capable of evaluating how well they have fulfilled these self-diagnosed needs by the end of the course" (Fowler, pg. 13). It would be ignorant of me, as an educator, to assume all adult learners are ready and capable of being entirely self-directed. Learners are at different stages in their transformation from being educational followers to educational directors of their own learning. Since most of today's students, young and old, were educated in pedagogical classrooms, it will take guidance, practice, and support for them to completely make the transformation and become entirely self-directed. I am experiencing this lack of learning maturity in my current teaching position, especially since I am using some methods (which I will discuss later in this paper) that were not exercised in their past traditional mathematics classrooms.

Pedagogy has been teacher centered in nature, where all attention is on the educator, whereas, andragogy is learner centered, focused on what is happening within the learner. Andragogy implies a shared concern for the learning rather than a pedagogical expectation that learning will take place no matter what. Building a partnership, between teacher and learner, and pulling for the common goal of learning is the nature of andragogy. The educator and learner need to have a connection, a personal relationship. The learner needs to know (1) you care, (2) that he is valued, (3) there is a sincere interest in their progress, and (4) you are there to support them.

Creating this support goes beyond the individual, but the class atmosphere must be supporting, trusting, and relaxing as well. I believe that I work at and am affective at establishing a supportive environment in which students can learn. The book cautions against making a class that is too safe and supportive, because this inhibits the student's choice to contend with each other about concepts and ideas. I firmly believe that concept struggles, either individually or as group, 'stick' better than those ideas taken at face value. Developing a relaxing atmosphere is something I could work harder at. It is difficult to balance between keeping up mental energy so students are ready for the next challenge and still having smooth transitions into each learning activity. Part of me says that a classroom where the mental energy is high can not be relaxing. Yet, when mental energy is high the room is bounding with excitement. On the other hand, the room should not be so electric that it intimidates the shy or less confident learner.

As I am sure many other educators struggle with time constraints, I find this is the greatest limiting factor to developing intimate relationships with students and creating a relaxing environment. It is assumed that strengthening teacher-student relationships would mean that I am able to personally interact with each learner at every class meeting, but where is the time? As a second note on relaxing class atmospheres, there is always more to do than time allows for. Everyone thinks, evaluates, explores and learns at their own pace. These things take time; they are not something that can be managed like a timed test.

People learn better and recall information that is structured and organized into related clusters. My job, as the instructor, is to help make those clusters bigger, building on old ones and bridging gaps between existing ones. In the practice of pedagogy information tends to be more structured because of the nature of pedagogy to rely on content rather than learner processes. This is where my strong use of pedagogy comes in. I plan everything. I can not just 'wing' it. I anticipate learner's responses, hang-ups, and have my own idea of how the process will take shape. Although these predictions are some of the foundations I was taught in my undergraduate education degree, I can see how this process focuses on what I believe they should learn rather than what they want to learn. In either case I do not deny that a greater amount of learning takes place when the class is well structured and organized in comparison to class in which random learning is explored without immediate connections. I believe this need to be organized is due to the educational experiences I have had. It is said that teachers teach how they have been taught.

In addition, "Studies have shown that teachers teach the way they learn" (Bradley, pg. 67). It is suggested throughout the book that instructors should examine their learning and teaching styles to better understand themselves as educators. I have observed many times where my teaching and learning has had a direct relationship to the teaching and learning experiences I have had as a student.

I would like to conclude my discussion of the instructor with a powerful quote from the book. "As a facilitator of learning I have to relinquish the authority and responsibility of the educational situation to the learner. I can not control learning. I can only nurture it. It is my job to challenge the learners at the right time in the right manner and then to support them in their attempts to find out what this world is all about" (Queior, pg. 29).

Pedagogy and Andragogy in Content (Writing vs. Mathematics)

Often people have anxiety when faced with writing tasks. They are not confident in their writing. They believe that others can produce clear and effective writing without any great effort. Very little planning is often a mistake made by people who exhibit fear of writing and consequently can not effectively organize the content of their writing. Anxiety frequently relates to bad experiences in writing courses. I too am a person that is extremely apprehensive about writing. I am not confident in my writing; therefore, it is something that I struggle with a great deal. Conversely, I do not recall ever having bad writing experiences. It is believed that people like myself should take classes on technical writing (pedagogy) rather than an essay based class (andragogy). It is also thought that people who have high anxiety should be in a supportive, trusting class where students evaluate, reflect, and provide feedback on each other's writing. "Adults must be empowered to look beyond their anxieties and preconceptions and evaluate, and allow others to evaluate, the product of what is, by its very nature, an intimate act" (Frayer, pg. 5). An additional factor to writing anxiety is the objectives and evaluations are usually very subjective, completely determined by the reader of the document.

In the same regard, math anxiety is high among learners. All to often students, like those who fear writing, are not confident, but can usually do well and equally match their peers. If they are not equals it is because they are in such fear of making a mistake that they second-guess themselves, not trusting their intuition and sense of reasoning. Similar to fearful writers, most math complications come from not having a plan. I teach, and demonstrate in most problems I work for the class, a 5-step model for problem solving. The planning and the process of successfully working math problems must be organized. Bad experiences in previous classes contribute to current math dilemmas as well. Math classes focused on problem solving and critical-thinking (pedagogy/andragogy) would be better for a high-anxiety student than a concept based class (pure andragogy). In my courses a large component of the curriculum is writing and communicating mathematics. All work, including notes, solved problems, and written work, are kept in a portfolio. I have students discuss, critique, reflect, and encourage the work in each other's portfolios. Objectives and evaluations are clearly defined in math courses. Usually it is black or white; either the answer is right or wrong. The thought of all or nothing contributes to the student's fear of doing mathematics. Students have found that I evaluate on content and reasoning (hence writing component) rather than solely on process and rules.

Through my many years of education I never really understood pedagogy. This is sad to say considering my undergraduate degree is in secondary education. Pedagogy was a term I had heard thrown around here and there, but it was never discussed in length. All I knew was that it was a word to describe the teaching of children, other than that pedagogy had been a mystery.

On the same note, andragogy has held the same mystery for the past year. I knew it was a word to describe the teaching of adults, but what it really meant was still unclear. In reading this book both pedagogy and andragogy were not only defined, but also clarified the differences, explained how they are used in the classroom, pointed out that they both benefit the learner, and examined the pros and the cons to both.

I am wondering why andragogy is used only to define adult education. Why can it not be pursued more actively in young adults, or even young children? It seems to me that andragogy is similar to that of the Montessori philosophy. Why isn't all education a strong balanced mixture between pedagogy and andragogy? Why do we not teach children the skills to be critical, reflective thinkers? I believe that all education is leaning this direction, but it is a slow process. We (adult educators) should be increasingly aware of public education and prepared to assist in the development of that education. Who would be more aware of the profound future needs and abilities of children than those who deal with them as adult learners? Remember the children of today are the adult learners of tomorrow.