**A Critical Analysis of the Teaching Practice of Roberta Lacefield**

In this critical analysis of my teaching practice, I will act as my own process consultant (PC) as I use Apps' model for conducting a critical examination. Schein gives an excellent portrayal of the role of a PC in his classic book, ________. Since I do not have an external PC, I will attempt to place myself outside of my experiences and serve as a virtual PC. In the virtual role of a PC, I will facilitate this critical examination and search for inconsistencies in my assumptions, definitions, metaphors, slogans, questions, and reasoning. In addition, in the role of PC, I will outline a plan for improving my practice.

The first step in the work of a PC is to determine the reason(s) the client has sought the help of the process consultant. In my case, I have taught in the Developmental Studies Division of Waycross College for five years. I feel that it is time to look critically at my practice, to decide whether the pedagogical direction I have chosen is appropriate, and to look to the future of my practice. I feel that the use of a virtual PC can assist me in this critical analysis. One of the most difficult aspects of any critical analysis of one's own practice is determining implicit assumptions. The use of a virtual PC may allow me to step outside of my skin and see my practice more clearly.

Stating explicit assumptions is the first step in Apps' model. In my case, determining explicit assumptions is relatively easy to do because I already have such a list of assumptions. Last term, I shared my assumptions with the students in my classes. My experiences in the adult education program at FSU caused me to realize that all instructors hold assumptions about the students in their classes and about the process of learning. Stating these assumptions allows the students to understand the situatedness of the instructor. Therefore, I had spent some time considering my assumptions. The assumptions I shared with the students in my class are:

A) Anyone who can learn a language can learn algebra since algebra is the language of mathematics.

B) You are here because you want to learn math. My job is to serve as your resource and will help you determine what you need to accomplish your task.

C) Group learning is important because:

1) Businesses value employees who can work with others (see US Dept. of Labor report.)

2) Research shows that students who work with other math students learn moreand stay on task longer.

D) Metacognition is a powerful tool for learning. One of the most important thing you can learn from this class is to understand how you learn mathematics.

These are the stated assumptions I shared with the students in my classes. Some assumptions I also believe I hold but which I did not state are:

E) My ultimate goals for the class are:

1) to teach students mathematics which they can use in their everyday life and to make

them aware of the mathematics inherent in their world.

2) to share with them my love of and enthusiasm for the subject.

3) to encourage them to be actively involved in their own learning and to understand

that you can learn more from mistakes than from successes.

F) I am a valued and appreciated employee of Waycross College.

I also hold some assumptions about the skills I teach.

G) The skills I believe are important are

1) to understand that the solution to an equation is the set of numbers that makes the original statement true and a graph is a visual representation of the solutions.

2) to know the order of operations.

3) to know how to use a calculator and when such use is most appropriate.

4) to have a process to solve problems which includes a combination of equations, tables, and graphs.

5) to understand that a formula is a description of a pattern and to be able to use a formula.

At the time I decided that these are my assumptions, I did not check to make sure that they are consistent with my implicit assumptions and my process as outlined in the syllabus, as required to pass tests, and as reinforced through my words and actions. As a virtual PC, it is my role to look past the surface and to seek evidence to support or refute these assumptions. Before such an examination of concrete examples of my assumptions occurs, it is important to go back to Apps' model and give a set of definitions for the terms I will be using. These definitions may also reveal discrepancies in my reasoning. For example, what does it mean to "incorporate the use of the graphing calculator?" This clearly could mean different things to different people. Until I determine what __I__ mean by incorporation, I won't be able to decide whether that level of incorporation is appropriate or whether I am working at that level in my practice. Clearly this term needs to be more clearly defined. This realization of flaws in reasoning caused by not examining assumptions definitions, slogans, and metaphors is an example of the purpose of a critical analysis. In the process of examination, questions about assumptions are raised.

**Definitions:**

__The Standards__ - mathematics education which incorporates the NCTM/AMATYC Standards. These standards include de-emphasis of paper-and-pencil drill, one-step problems, cookbook problem-solving, and contrived equations and increased attention on active learning, integration of mathematics topics, and appropriate use of technology for computational work and graphing. I consider myself progressive in my use of the standards as a model of teaching. I look at my syllabus and choice of assignments supports the idea that I am integrating topics and using technology. However, my tests involve many one-step problems and cookbook problem-solving.

__NCTM__ - National Council of Teachers of Mathematics: a group primarily interested in K-12 mathematics.

__AMATYC__ - American Mathematical Association of Two-Year Colleges.

__active learning__ - any learning which is not passive but instead requires mental or physical involvement on the part of the learner. Examples include discussing material outside of class, reorganizing notes, and engaging in additional research beyond the requirements. Research supports the idea that it is through active learning that material is understood and retained. Thus, I should be providing opportunities and resources for students to engage in active learning.

__lab__ - an active learning approach which involves the learner in activities which lead to the discovery of concepts. Labs tend to be application based. Because of this, I believe they are a good way to teach adults. A glance at my lesson plans show that despite this belief, I only occasionally provide lab experiences.

__ASC__ - Academic Support Center: This is a center which contains resources such as tutors, videotapes, computers, additional texts, solutions, etc. I encourage, but do not require, students in my classes to use the center. If active learning is important, why am I not requiring students to use the ASC? I am a resource, why am I not providing leadership in steering students toward these tools they usually need.

__reform mathematics__ - that mathematics which is described by the AMATYC standards (see above.) The question is, is that the mathematics I teach?

__process education__ - a teaching practice which includes cooperative learning, guided discovery approach, critical thinking, problem solving, journal writing, and self-assessment. Since my experience class-testing a process education text, I believe in the efficacy of this approach. However, my syllabus does not reflect this belief since it shows no opportunities for journal writing and self-assessment and no reward for engaging in cooperative learning. This, despite the fact that my stated assumptions include a reference to the importance of group learning.

__mathematics__ - The definition of mathematics which I use is that mathematics is the study of patterns; algebra is the language used to describe and analyze patterns. This definition reflects the relevance of mathematics across disciplines since patterns occur in all fields.

__Developmental Studies (DS)__ - a series of courses designed to prepare underprepared students for college courses. Its use as a description of the program at Waycross College is consistent with Boyle's definition since the programs primary goal is to solve an individual, group, or community problem and the objectives are (ideally) developed out of the needs of the clients.

__intermediate algebra__ - content which is approximately equivalent to that attained in the second year of algebra at the high school level but also includes experiences necessary to understand these concepts. As I say this, I realize that I rarely provide such experiences.

__student evaluations__ - evaluations done once each year by the students about the instructor and the course. They are a good indicator of student perceptions. I generally score well in sharing my enthusiasm for the subject and improving student attitudes, two of my explicit goals.

Once definitions have been determined, the next step for the virtual PC is to look at slogans and metaphors at work within my practice. Like the definitions I choose, metaphors can provide evidence of my implicit assumptions. The metaphors I use expose my true assumptions. Metaphors are very powerful because they are subtle. They are a type of cognition in which the identifying qualities of one thing are transferred in an instantaneous, almost unconscious flash of insight to some other thing that, is by remoteness or complexity, unknown to us. This is an "aha" kind of knowledge that states more than our words. It may also state things we do not intend. The following is a list of my most common metaphors and the context in which I generally use them:

"We're not building anything; we're just hammering." - When we are working on skills rather than integrated topics, I will often say this so that students understand there is no big picture involved. I regularly use this metaphor to explain the bigger picture in the Math 098 (Beginning Algebra) class. Unlike Math 099 (Intermediate Algebra) it is predominately a skills based course in which we never "build" anything; we simply hammer. The obvious question is what this metaphor of a mindless, repetitive action says about my own feelings about the class.

"Well, I'll stop preaching now." - I tend to preach when I suspect students have not done the preliminary work they need to do in order to understand a concept. The metaphor "preach" shows that it is a path I would like them to take but one that I know they often don't.

"No pain, no gain." - generally said when I am preaching about homework. the implicit assumption is that if the student is not struggling, learning is not occurring.

"attack a problem" - my terminology for undertaking a difficult problem. This metaphor clearly indicates that a problem is something of which the student should be wary.

"my students" - a way of talking about students in my classes. I am trying to stop using this term since I do not believe it is appropriate for adult students.

"brain as hard drive" - I often use the analogy of floppy drives, saving to the hard drive, organizing files, etc. when discussing memory.

In addition to the metaphors I use, as a department we use many slogans. The slogans can be useful in determining the implicit assumptions inherent in the environment.

"We are an open admissions institution." - This statement implies an inclusiveness which has little to do with reality. Although anyone can attend the college, anyone can also flunk out. Without a certain level of experience, it is highly likely a student will.

"They don't do enough homework." - This oft-repeated explanation for student failure assumes doing the homework is necessary and sufficient for success.

"They need to examine their priorities." - The assumption is that students have a choice in their priorities. Often, this is not the case. Children and jobs simply must come before classes.

"Waycross College: Leading the way into the 21st Century." - Clearly, the question raised by this slogan is whether it is the truth. Within my practice, I need to decide whether I am using the current research and whether my practice is preparing students for the mathematical issues they will face in the future.

An examination of my definitions, slogans and metaphors all begin to give indications of my implicit assumptions and any contrast with my explicit assumptions. Some of the questions which come out of this contrast are the following: Is my curriculum a reform curriculum or is it actually a traditional curriculum? Do my tests require knowledge of the skills and ideas I say I believe are important? Do I incorporate metacognition into the curriculum? Do I use the research to improve my courses or do I merely read the research? What behaviors does my syllabus encourage? discourage? Are these the behaviors intended?

In the process of answering these questions or addressing the issues raised by them, it is important to examine my reasoning. One assumption which must be addressed is that changes I might make to my practice are an improvement to the program as a whole. In addition, as a PC I must look at the environment in which such changes will occur and determine their effect on the larger organization. When I looked at my personnel folder as a part of this analysis, I was surprised by the content. I did not find references to grants I had written, presentations I had done, workshops and conferences that I had attended, or committees I had chaired. Instead, I found references to two instances when I had followed the incorrect chain of command and one negative response to a grant I had written. This is not what I expected to find. It was clear that while I believed my efforts on behalf of the college were valued, my personnel file seemed to show a misconception of what these efforts might be . Thus, any remedies, changes, or improvements I encourage in my role as virtual PC must also be considered within their context and the negative perception they might garner.

Although Apps' model ends here, as a PC my work is just beginning. After considerable thought, self-examination, and internal discussion with my virtual PC, I have come up with the following remedies, changes, and improvements. I hope these will address the questions, shortcomings, and lapses in reasoning found in my critical analysis.

1. Incorporate an additional alternative form of assessment each quarter/semester for the next four quarter/semesters. These might include such forms of alternative assessment as journal writing, group projects, take-home test questions, individual projects, and portfolios.

2. Decrease the amount of lecture time. To aid in assessment of achievement of this objective, I will record the amount of time I am spending lecturing.

3. Increase incorporation of metacognition and active learning. Begin this process by asking for the help of student life in determining students' personality types. Continue the incorporation by actively seeking opportunities to tap into Gardner's multiple intelligences.

4. Keep a personal teaching journal. After all, sharing failures and ideas is a part of scholarly work.

5. Address the perception outside the department of the changes I am making. Be aware that such changes can be misunderstood.

This list represents a large undertaking and a powerful change from my implicit assumptions. However, I feel that using this list will address some of the inequities I have found between my explicit and implicit assumptions. In addition, I feel this list gives me the direction I was looking for when I began my work at FSU. However, I must be aware that additional analysis will be required and probably shouldn't wait until I complete another five years at Waycross College!