DEPARTMENT OF MATHEMATICS EDUCATION

Preliminary Examination
Doctor of Philosophy

Prepared for:
Dr. Shawn Glynn

				
Lisa Sheehy
July 1996

										
In response to:

Question #1: All mathematics teachers want their students to learn "meaningfully." (In the Standards, the term "conceptual" is used; and Skemp uses the term "relational.").  In your opinion, what exactly is "meaningful mathematics learning?" And, in particular,...
	•	How exactly, should a high school mathematics teacher promote it in his or 	her students?
	•	How exactly, should a high school mathematics teacher assess it?
	•	What roles should cooperation and competition play in the meaningful   	 	learning of mathematics?
	•	What are the cognitive consequences (i.e., achievement) and affective 	consequences (i.e., attitudes) of cooperation and competition?

In your answers, be sure to include concrete examples.

Meaningful Mathematics Learning

	Although it may seem that the current reform efforts in mathematics education are the first to seriously address the development of meaningful mathematics instruction and learning, we can, in fact, trace the history of mathematics education in this country ( Stanic & Kilpatrick, 1992 and Kilpatrick & Stanic, 1995) and observe professional mathematics educators analyzing and discussing many of the same questions current educators are addressing:
	1.  What mathematics do we teach? and why?	
	2,  How do students learn mathematics?
	3.  What is the best way to teach mathematics?
I believe at the heart of these decisions lies the more comprehensive question "What is meaningful mathematics learning?"  This is the question that I am not only to address in this exam, but also the one that drives my decisions as a high school mathematics instructor.
	In the following discussion of meaningful mathematics learning, I will initially offer a range of ideas and definitions of meaning and a variety of learning theories with respect to meaning.  Next, I will state my  present beliefs about meaningful learning.  Finally, I will  describe what a mathematics classroom should look like from my perspective of what meaningful learning is for high school mathematics students.  

The Meaning of Meaning	
	It appears that the educators who come after tend to accuse the ones before of not teaching mathematics "meaningfully."  In 1945, William Brownell offered the following discourse on the meaningful teaching of arithmetic:
Nowadays it is fashionable to say that arithmetic should be taught meaningfully.  It was not always so.  Indeed, it was not so short a twenty years ago, when a few students of the subject began the campaign to make arithmetic meaningful.  At that time, those who talked about meaningful arithmetic were regarded as perhaps deluded and certainly unrealistic.  But times have changed....And publication of each new  or revised textbook is making arithmetic meaningful.  There is, then, general agreement at present that arithmetic should be taught meaningfully.  This agreement in theory is not, however, matched by agreement in practice (p. 481).
Although we may find humorous the similarities of Brownell's discussion to today's commentaries on educational issues in mathematics (i.e. the use of technology, alternative assessment techniques), we must look deeper to determine the source of such recurrent arguments.  	
	I believe that it is not the decision to teach mathematics meaningfully that drives educational reform efforts, but the manner in which we interpret the term meaningful mathematics learning that has been and will continue to be the force behind calls for change in mathematics education.  Of course mathematics educators have always desired effective and meaningful instruction.  It is certainly a ludicrous argument to say the opposite.  The desire to provide effective education to students has not changed.  What has changed over time is the interpretation of what it means to teach and learn mathematics meaningfully.
	Thus, to describe meaningful mathematics learning  "exactly" would be to try to place parameters on three terms in the field of mathematics education that are not only (and in my opinion, should be) very dynamic in nature, but are also constant sources of research, discussion, and debate.  Although there may not be universal agreement among mathematics educators as to the exact definitions of the above terms, I firmly believe it is crucial to effective instruction that each individual educator define his or her own convictions and beliefs about meaning, learning, and mathematics.   So how does one arrive at a personal definition of learning meaningfully? It is important for an educator to carefully examine the ideas and research of those committed to defining the terms, weigh those ideas against personal experiences, develop a personal framework and philosophy for the teaching of mathematics, and then plan instruction accordingly.  This process is one that I plan to emphasize as I write this exam. 
	Roget's Thesaurus offers as two synonyms of meaning both interpretation and sense.  As I discuss meaning from the perspective of a variety of philosophers, psychologists and mathematics educators,  remember that I am only interpreting other's sense of the "meaning of meaning,"    
•	For Plato (1966), perhaps meaning was synonymous with "judgment."  He stated 	I have a notion that, when the mind is thinking, it is simply talking 
		to itself, asking questions and answering them, and saying yes or no.  
		When it reaches a decision-which may come slowly or in a sudden 
		rush-when doubt is over, and two voices affirm the same thing, then 
		we call that its 'judgment.'   (cited in )
•	The above view of equating some type of internal affirmation with meaning is echoed much later by John Dewey.  Dewey (1933) in How We Think defined meaning in relation to understanding.  He stated "to understand is to grasp meaning." (p.132)  Dewey felt that meaning actually developed in three stages.  We are triggered by curiosity and moved to inquire.  At this time meaning is an idea.  Meaning is now conditionally  accepted for use and trial.  When it is positively  accepted, some object or event is understood, thus meaning is grasped (p.133).
•    Ausubel (1968) places the emphasis on existing knowledge.  He believes that meaning occurs if "the learning task is related in a non arbitrary fashion to the learner's existing knowledge" (p.126).  He contrasts meaningful learning with what he calls rote learning.
•	Glynn and Duit (1995) offer a similar view of meaning that develops from the relationship formed between existing knowledge and new experiences.  In a discussion of learning science meaningfully, they state "students learn...meaningfully when they activate their existing knowledge, relate it to educational experiences, and construct new knowledge in the form of conceptual models." (p.5)
•	In an interpretation of Piaget's belief that meaning is derived from language, von Glasersfeld (1991) interprets meaning as re-presentation.   He defines two types of meaning as follows:
		First,  figurative meanings are those that can be visualized 
		immediately because they call up a re-presentation of the kind 
		of sensorimotor experience from which they were abstracted.  
		Second, operative meanings cannot be re-presented as such but 
		only as 'implemented' in a sensiormotor situation. (p.98)
•	In line with Piaget's emphasis on language in meaning development, Steffe and Cobb (1988) define meaning as "whatever aspects of one or more schemas that are associated with a word or words within the awareness of the child" (p.337).  They define a scheme in three parts:  "first is the child's recognition of an experiential situation as one that has been experienced before,; second is the activity the child has come to associate with the situation; and third, is the result the child has come to expect with the activity "( p.337).
•	Placing increased emphasis on social interaction, Blumer (1969) states "meaning  of a thing for a person grows out of the ways in which other persons act toward the person with regard to the thing...Thus, symbolic interactionism sees meaning as social products, as creations that are formed in and through the defining activities of people as they interact" (pp. 4-5).   Yackel (1995) interprets Blumer's view of meaning in the statement "meaning is not intrinsic but develops out of interaction and interpretation" (p.134).
•	Cobb (1995) reiterates this social view of meaning with the term taken-as-shared meanings.   Cobb states "children's cognitive constructions occur as they contribute the classroom community's establishment of taken-as-shared mathematical meanings and practices" (p.37).
•	A much stronger view of the role that social interaction plays in meaning making can be seen in Konold's (1995) interpretation of Gergen:  "Gergen sees knowledge not as something that resides in individuals at all, but in a text, or more precisely in the collective status of texts.  Knowledge belongs to the social realm because the knowledge status of some proposition (or text) is not determined by any one individual, but collectively by all individuals in a position to render judgment."
•	In van Oers' (1996) interpretation of  Leont'ev we have to distinguish cultural meaning from personal meaning.	
Cultural meaning refers to the generalize knowledge and skills 
for dealing with the world that have been built up throughout 
cultural history.  Broadly speaking we can conceive of cultural 
meaning as a body of cultural knowledge (including both procedural 
and declarative knowledge), embodied in the historically approved 
and symbolically codified actions and methods of acting in a certain
 domain of human activity.   Personal meaning (sense) is based on 
the attribution of personal valued to the actions and goals as actually 
generated within an activity from the perspective of one's personal 
motives.  (pp. 93-94)
•	Skemp (1976) also offers two types of meaning (understanding) categorized as instrumental and relational understanding.  He defines them as follows:
		 Instrumental Understanding:  knowing what to do ('rules without reason')
		Relational Understanding:  knowing what to do and why (p.10)
	Skemp believes that both types of understanding equate meaning.
•	The National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards  (1989) parallel the ideas of Skemp by using the terms procedural and conceptual knowledge throughout their recommendations for school mathematics programs.   An example will help clarify the differences.
	Consider the problem of adding 1/3 and 1/2.  A student with strictly 	procedural knowledge would be able to calculate the correct answer by :
		1.  Calculating the common denominator...       6
		2.  Calculating equivalent fractions...             	 2/6 + 3/6
		3.  Adding  numerators...				5/6	
	A student with conceptual knowledge of adding fractions with unlike 	denominators would be able to answer the following questions:
		1.  Why do we need a common denominator to add fractions?
		2.  What does it mean for fractions to be equivalent?
		3.  Why are 1/3 +1/2 = 2/5  and  1/3 +1/2 = 5/12  incorrect statements?
NCTM suggests that true meaning only occurs when conceptual knowledge is      constructed by the student.
	Certainly there are differences in the ways in which "meaningful" has been defined or interpreted.  However, whether the focus is on individual development, social interaction, judgment making, activation of existing knowledge, skill acquisition, or conceptual understanding, there does appear to be agreement that meaning stems from our experiences and the sense we make of them.  I would argue then that the essence of meaning is experience.  So, what does all this mean for learning?

Meaningful Learning
	I believe any individual's theory of learning stems from an initial decision about how he or she believes students make meaning of their experiences.  At the core of every learning theory lies a fundamental belief about what defines meaningful learning  and the manner in which it occurs or develops (see Figure 1).  

		Figure 1:  Meaning at the core of learning theories

	Conditioning, with its emphasis on stimulus and response, was one of the first psychological theories carried over to human learning and, according to Fehr (1953), "still either consciously or unconsciously guides the teaching patterns in many of our classes" (p.11).  Conditioning as a theory of learning stems from the Behaviorist branch of psychology (i.e. Skinner).  I believe that for supporters of this conditioning theory of learning, meaning lies in the stimulus.  The goal of the teacher is to find the appropriate stimuli to produce a desired behavior in a student.  
	According to Fehr (1953), "The fundamental characteristic of connectionism is the bond established between a situation and the response made by the organism" (p. 13).  Thus, for the connectionists, meaning is found in the bond:  the stronger the bond, the more meaningful the learning.  Thorndike (1924) offers the following guidelines for connectionist teachers:
1.  Consider the situation the pupil faces.
2.  Consider the response you wish to connect with it.
3.  form the bond; do not expect it to come by a miracle.
4.  Other things being equal, form no bond that will have to be 
      broken.  (cited in Brownell, 1948, p.483)
	Contrasting the belief that meaning is found in the individual elements of a situation, we find the Field Theorists (i.e.  Lewin, Tollman).  One of the fundamental principles of this theory is that "the whole is greater than the sum of its parts."  When the various elements of a situation are grasped in their relation to the whole situation, insight occurs (Fehr, 1953, p. 19).  It is this insight that leads to organizing a "relationship of relations" into  an organized system of knowledge.  Thus, for a field theorist, meaning lies in this organized system of knowledge, not in isolated facts or skills.  According to Fehr, a teacher with this belief of meaning would encourage students to analyze, synthesize and deduce, until a system is built that can be drawn upon in new experiences. 
	Davydov's view of Soviet Activity Theory, as interpreted by Cobb (1996) is grounded in the idea that "formal,  abstract understanding should precede pragmatics problem solving" (p. 9).  The meaning in this theory is the ability to understand the abstract nature of concept or skill.  "Davydov notes that his instructional proposals involve an 'ascent from the abstract to the concrete' "(cited in Cobb, 1996, p.9).   Thus, a teacher would first offer the abstract concept of say multiplication to students and then have students apply the concept to concrete problems or applications.
	In sharp contrast to Davydov's view of activity theory are the theories supported by  inquiry  or discovery based learning.  Both of these views of learning find meaning in the generalization of concrete examples to a hypothesis of a more abstract concept:  a much more inductive approach to meaningful learning than those previously discussed.  Inquiry based learning differs slightly form discovery learning in that true inquiry based learning is driven by student selected problems, interests, questions, while discovery learning is what some critics may call teacher contrived learning experiences.   While both support active student investigation and construction of knowledge, teachers who believe that intrinsic motivation (i.e. an interest in solving a problem)  has a strong relationship to meaning making would most likely promote inquiry based learning.  Those teachers who believe that the thrill of discovering a concept is in itself motivation, may be more inclined to purposefully select investigations leading to certain desired results.  
	And then there is constructivism... Constructivists, whether radical, weak or social draw their inspiration from Piaget, for whom the individual is the central element in meaning-making (Cobb, 1994).   Ernest (1994) states "Piaget's constructivism led to the currently fashionable radical constructivist theory of learning mathematics" (p.62).  Cobb et al. state "from the constructivist perspective, mathematical learning is not a process of internalizing carefully packaged knowledge but it is instead a matter of reorganizing activity, where activity is interpreted to include conceptual activity or thought" (cited in Koehler & Grouws, 1992, p. 118).  Constructivism is a theory of knowing.  It is based in the fundamental idea that the only reality or truth we can know exists solely in the mind of the individual.  That is, individuals construct their own meaning:  meaning is in the mind of the individual.  In a truly radical sense of constructivism, the teacher can only make a model of what meaning might be for a student.  If we take this theory of knowing to the mathematics classroom, mathematics students become active in the processes of investigating, hypothesizing, discovering, concluding, reflecting, and adjusting.  In a constructivist classroom, it becomes increasingly important for teachers to be aware of students' meaning making processes and to build models of their students' knowledge.  Teachers use mathematical activities, observation, and dialog with students to begin to develop models of their students mathematical meaning.  According to Cobb and Steffe (1983), "in the constructivist view, teachers should continually make a conscious attempt  to 'see' both their own and the children's actions from the  children's point of view" (p.85).  
	There are researchers (Cobb, Lerman, Wertsch, etc.) who feel that drawing inspiration solely from Piaget results with too much emphasis on the individual and not enough on the social aspects of knowledge acquisition.  As a modified approach to a theory of knowing, these researchers are looking to the ideas of Vygotsky to develop the field of social constructivism.  As a theoretical foundation, they use Vygotsky's general genetic law of cultural development   which states:  "Any function in the child's cultural development appears twice, or on two planes .  First it appears on the social plane, and then on the psychological plane,.  First it appears between people as an interpsychological category, and then within the child as an intrapsychological category...Social relations or relations among people genetically underlie all higher functions and their relationships"  (cited in Wertsch and Toma, 1995).  While still rooted in the idea that meaning is constructed by the individual, the social constructivists place much more emphasis on affects of social interactions in this meaning-making.
	The analysis of the above learning theories with respect to meaning further supports the notion that the place and purpose of meaning is at the core of a theory.  Whether an educator constructs his or her own learning theory, adopts or modifies an existing learning theory, or combines several theories, it is the initial personal definition of meaning that guides the choice and ultimately determines the culture and organization of the classroom.  Thus, it is crucial to meaningful  mathematics learning that all mathematics educators come to know  as clearly as possible what they personally believe first about meaning, then learning, and then how these beliefs affect the teaching of mathematics in their classrooms.  

Meaningful Mathematics Learning 
	So back to the question,  "In your opinion, what exactly is 'meaningful mathematics learning?'"  I must start first with what is meaning?  Meaning is the sense that we make of our experiences.  Thus, meaning must be initiated by an experience in which the individual desires to make sense.   Meaning is internal, but stems from social interactions.   True meaning is conceptual in nature and is constructed by the individual.  Meaning is obtained when an individual can communicate conceptual understanding to another person or when a concept can be applied to a new situation.
	In order to define meaningful learning, Glynn and Duit (1995) state five conditions for learning science meaningfully.  I am borrowing this idea of stating conditions and adapting it to my beliefs about learning meaningfully.  For me, meaningful learning occurs when an individual...
	1.  is in a situation he or she desires or needs to make sense of
	2.  actively investigates, discusses, applies problem solving strategies to, 	  	     and/or studies aspects of the situation relevant to sense making
	3.  constructs conceptual understanding of the situation
	4.  communicates this understanding or applies it to another situation
I believe that, while meaning is internal it is obtained by the individual interacting with the external experiences. Thus, the process of meaningful learning is well facilitated by peer interaction and collaboration.  In summary, I align myself with Vygotsky in terms of the development of meaning and NCTM in terms of the importance of conceptual understanding.  I believe meaningful learning occurs when it is first experienced on the social plane (outside of the child) and then on the psychological plane (internally).  

Promoting Meaningful Mathematics Learning in the Classroom
"If the heart of instruction is the creation of intermental functioning that will maximally facilitate the kind of intramental functioning we wish to see, then it is necessary to identify ways to create an institutionally sanctioned 'space' for this kind of intermental functioning to occur" (Wertsch and Toma, 1995).
Wow!!  I know that is not a very formal way of expressing my feelings about the above conditional statement,  but that is exactly how I felt when I read it for the first time.  Those forty-five words have helped me define and make personal meaning of my responsibility as a classroom teacher.   It is my job to help "create an institutionally sanctioned 'space' for this kind of intermental functioning to occur."
For me that "space" is a cooperative/collaborative learning environment.   I view cooperative learning through the lens of the  social constructivist theory of learning.  Thus, cooperative learning is more than just a pedagogical tool, it is a personal philosophy about the limbering and meaning making of my students.  It is the framework around which I plan instruction.  So, an expanded version of Figure 1 for me would look like:

Figure 2
In my model,  meaning is defined as movement from the intermental plane to the intramental plane, learning theory is rooted in a social constructivist framework, and instructional planning is geared around the incorporation of cooperative learning strategies.
	So what is cooperative learning?  In order to best define cooperative learning, it may prove helpful to first determine what cooperative learning is not.  According to Johnson and Johnson(1994), putting students into  groups to learn is not necessarily the same thing as structuring cooperation among students.  They summarize that cooperation is not:
	1.  Having students sit side by side at the same table and talk with other as 	they do their individual assignments.
	2.  Having students do a task individually with instruction that the ones who 	finished first are to help the slower students.
	3.  Assigning a report to a group where one student does all the work and 	others put their name on  it.
So what is cooperative learning?  Kluge (1990) defines cooperative learning  "as a method of classroom instruction in which students are placed in small groups and wok together to achieve a common goal" (p.1).   According to Johnson and Johnson (1991), cooperative learning is based on the following assumptions
1.  Students can learn from each other.
2.  Students can, with instruction, govern themselves.
3.  The teacher is not the only source of information.
	Again, cooperative learning is not just a pedagogical tool, it is a belief about learning.  Students do learn from each other.   Social experiences do play a part in  meaning-making and the development of knowledge.  I believe that only when both the teacher and the students have a deeper understanding of what cooperative learning is and how to use or participate in it,  will cooperative learning become an invaluable tool for the classroom and beyond. 
	 John Owens summarizes that "the constructivist hypothesis states that one comes to know by an adaptive process of organizing one's experiences rather than by perceiving some external reality.  The Curriculum and Evaluation Standards for School Mathematics explicitly support this hypothesis:
In many classrooms, learning is conceived of as a process in which students passively absorb information, storing it in easily retrievable fragments as a result of repeated practice and reinforcement.  Research findings indicate...that learning does not occur by passive absorption alone.  Instead, in many situations individuals approach a new task with prior knowledge. assimilate new information, and construct their own meanings...This constructive, active view of the learning process must be reflected in the way much of mathematics is taught."  (cited in Penderson and Digby, 1995)
	 The use of cooperative learning in the classroom relates to and supports the Standards proposed by NCTM.  First and foremost, cooperative learning supports the standard of Mathematics as Communication.  Cooperative learning activities rely heavily on students abilities to put their mathematical ideas into words and express them to their team members.  As students learn to communicate better mathematically,  cooperative learning improves and vice versa.  Cooperative learning also helps to foster an environment where students are encouraged to take risks and explore their own ideas.  Self-confidence is improved as students become willing to share ideas with their team members.  In my opinion,  the main objective of the NCTM Standards is that students become active participants in the learning of mathematics that is meaningful to them.  A student who is a part of cooperative team, becomes active not only in his own learning but in the learning of others.  Since he is creating and discussing the mathematics,  it becomes a meaningful activity. 
	Mathematically, a goal that I have for my students is that each one develops mathematical power.   NCTM (1989) defines mathematical power as the ability of a student  to explore,  conjecture, and reason logically, as well as the ability to use a variety of methods to effectively solve non routine problems.  I define mathematical power in the words I most like to hear from my students... "Don't give me anything more than the problem.  I want to figure it out by myself or with my team."  I feel that as a secondary mathematics teacher, one of my primary responsibilities is to help bring my students to a place where they not only feel confident in their abilities to solve mathematical problems, but they also have the desire to explore the "unknown."   Giving students the tools with which to solve mathematical problems is only half the battle in developing mathematical power.  The other half of the battle lies in fostering a sense of curiosity in students.  It must become not only important but necessary for students to ask why , to not accept anything for truth until they have proved it to themselves, and to make and explain their own conjectures about problems or situations.  Until this desire and curiosity are a part of a student's learning process, all the mathematical ability in the world will not give him, what is in my opinion, true mathematical power:  a desire to solve a problem or explore a situation and then the confidence in his own abilities to begin the process. 	
	My goals for instruction are then to provide my students with problem situations that are instrumental in 
	1.  developing conceptual understanding about mathematics
	2.  fostering the development of mathematical power
	3.  helping students use social interaction to make leaning more meaningful.
The following lesson is one that strives towards the attainment of the above goals.
Note:  The type of cooperative learning to take place in a classroom relies heavily on the amount of experience students have had in a cooperative situation.  Students must learn  about cooperative learning before they learn through cooperative learning.  I suggest that a teacher spend a few days just discussing with students  the following concerns:
	What does it means to work with others and be on a team?
	How or when do you need to assign roles? 
	What is appropriate conversation in a team? 
	What are the benefits of being in a team?
	What are some problems that may arise?    
	How can I be encouraging to others on my team?
Only after cooperative learning has been discussed and concerns have been addressed should cooperative activities be introduced.  The following activity is a good one for students without much experience in cooperative learning.

ACTIVITY 
Roles:  You will need to be in a team of four.  Each team member will accept one of the following roles.  Discuss these roles and how they might function best in your team and then assign them accordingly.
	1) Leader/Encourager
	2) Technology Expert
	3) Records Keeper
	4) Presentation Coordinator
Materials:   graphing calculator
	         overhead pens and transparencies
	         pencil and paper
Task:   Use your graphing calculators to examine the function f(x) = a(x + b)2 + c.  Make conjectures about  how the values a, b, and c will affect the graph of the function individually and in relation to each other.  Organize your findings and/or conjectures.  Create a presentation based on your discoveries.
Teacher Role:  It is the role of the teacher throughout this activity to help students learn to work together on the task and to encourage each student to constantly be aware of his or her own role as well as the goal of the team. Teachers must remember to encourage students to participate in and be actively involved in all team activity. 

Cognitive and affective consequences of cooperation and competition
	Johnson & Johnson (1991) classify the structure of general classroom instruction in three main forms:  competitive,  cooperative, and individualistic.  They claim that "for the past fifty years competitive and individualistic efforts have dominated classrooms" (p. 26) and that "cooperative learning has been relatively ignored and underutilized by teachers even though it is by far the most important and powerful way to structure learning situations" (p. 26).   Research continues to find that cooperative learning is an effective method of teaching if it is clearly understood by the teacher and used in a manner that parallels the philosophies underlying it.  If teachers understand cooperative learning and carry it out in their classrooms as suggested by researchers, the teacher may find that cooperative learning tends to promote higher achievement, the greater use of reasoning strategies, more positive relationships between students, more positive attitudes toward subject matter, and higher self-esteem (Rottier, 1991, p. 16).  Research has shown that "because of the quantity of cognitive rehearsal, all students of all ability levels in cooperative learning groups enhance their short and long-term memory as well as their critical thinking skills and that because cooperative learning leads to positive interaction among students, intrinsic learning motivation and emotional involvement in learning are developed to a higher degree" (Bellanca, 1991, p. 242).  In terms of assessment and grading, Slavin et al (1985) found that "joint grades and rewards seemed to be perceived as fairer by students than the traditional competitive and individualistic grading system."  Cooperative learning also proves to have favorable results in the area of ethnicity and cultural issues in the classroom.  Kluge (1990) summarizes that "in some cases, cooperative learning was found to be more likely to increase the academic achievement of non-white students than that of white students" and that "liking among students of different races increases when cooperative learning methods are used" (p. 12)

Assessing Meaningful Mathematics Learning
	  I believe that they key factor in appropriate, fair and quality assessment lies in the fact that assessment  is to be a direct reflection of what is happening in the classroom.  In fostering meaningful mathematics learning through lessons and activities, I ask my students to analyze problem situations, create conjectures,  explore those conjectures, and communicate their findings or ideas.  The assessment of mathematical learning should provide these same opportunities to my students.  The authors of Assessing What Counts  (1993) make the following statements about mathematics assessment

"Rather than forcing mathematics to fit assessment, assessment must be tailored to the mathematics that is important to learn."

"Assessment should be seen as an integral part of learning and   teaching rather than as the culmination of the process."

"The best way to provide opportunities for the construction of mathematical knowledge is through assessment tasks that resemble learning tasks in that they promote strategies such as analyzing data, drawing contrasts, and making connections."
These statements support and illustrate my personal beliefs about appropriate, ongoing, meaningful assessment in the mathematics classroom.  The following lesson is an example of how I might assess the mathematics learning of a student :

ACTIVITY
Materials needed:  A set of 1" by 1" squares, paper, pencil, calculator, blank overhead transparencies, and overhead pens
Activity:  The task of your team today is to explore all the  different dimensions of rectangles that can be constructed using  specific numbers of your square manipulatives.  
	Example:  You  can only make one type of rectangle (dimension 1 x 2) with  	two squares.   
How many can be made with three squares? four squares? etc.  I want you to investigate this situation with your team.  Please keep a record of the data your team collects.
Discussion Questions: ( to be given to the teams after they have investigated the problem for a while - I don't want my questions to guide their explorations)
	1)  Tell me about any and all  interesting patterns that your team has found.
	2) Can you find and discuss any mathematical reasons why these patterns 	would be occurring?
	3)  Can you think of ways to categorize different numbers of squares based on 	their ability to form one or more rectangles?
Please take out your journals at this point and do the following entry individually.
 Journal Entry:  Tell me what you and your team have been working on today.  Explain how your team approached the investigation.  What do you think that you personally contributed to this investigation?  Try to briefly summarize the conjectures your team made about this activity.  Bring me your journal when you have finished this entry and return to your team to continue the investigation.
Presentation:  I would like your team to  now develop a presentation of your  mathematical findings and/or conjectures to the class.  You will all need to have a part in your team's presentation.   Please give me your team's discussion questions at the end of your presentation.    
Assessment:  This activity will be assessed in four ways.  First , each student will be assessed on his or her involvement in the team's activity.   Second, the student's team will be assessed on their explorations based on the discussion questions that they work on and turn in.   Third, the student will be assessed on his or her individual perception of the activity, conceptual understanding,  and ability to communicate that perception in a written form (the meaning constructed so far) .    Fourth, each student will be assessed on the presentation.   In assessing the student in these four areas (cooperative skills, investigations, conceptual understanding, and communication),  I feel I will send the students the message that the entire process of learning is important, not just the final results. 
Conclusion
	In conclusion, my definition of  meaningful mathematics learning stems from decisions about meaning, about learning, and about how this plays out in my classroom in terms of instruction and assessment.  I believe that constructing a personal definition  of meaningful mathematics leaning in this manner is a process that every teacher should go through.  The more clear a teacher is on these personal decisions, the more effective  instruction will be.  That is what all educators want...to give their students the most effective and meaningful instruction possible.  	
"If we as teachers hope to encourage critical thought in others, we must engage in it ourselves.  Throughout out teaching careers we must participate in an ongoing, collaborative process of reevaluation of , and liberation from, our taken-for-granted views." (Berlak & Berlak, 1987, p.170) 

In response to:

Question #2:  Draw your own model of mathematics instruction.  (Most

 models can be represented as diagrams, so I imagine yours can be, too.)  

Your model should serve as a guide for planning lessons and fostering 

meaningful learning in high school mathematics classes.  It should contain 

most of the major components that mathematics educators would agree 

are essential elements of effective lessons.   Justify the elements you select 

by writing a paragraph about each of them with appropriate references to 

the mathematics education literature.

PUT THE MODEL IN HERE!!!

Meaningful Mathematics Lessons:  
A Model for Instructional Planning

	The model of instruction I have developed, parallels the ideas that I put forth in response to Question 1.  It represents the process that I believe teachers should go through as they plan for instruction.  I do not believe this process is a "one time deal."  This model presents questions that an educator should be asking oneself constantly.  I stated earlier that the terms meaning, learning, and mathematics are dynamic in nature.  Ten years form now my view of learning will probably be different than it is now.  Experience, new research, studies, communication with other educators...all of these aspects of education will foster my continual growth professional as an educator and my personal beliefs about education.  While I expect the process to be continual, I feel it is extremely important to effective lesson planning an educator is clear about what one currently values in the mathematics classroom and believes about one's self, and one's students before lessons are planned.  
	Thus, this model is to be read from left to right.  I have presented what I consider to be the major components essential for meaningful mathematics instruction.  The model is from the perspective of an individual classroom teacher.  It is presented as a series of questions that result in a lesson that promotes meaningful mathematics learning according to the teacher's personal beliefs about meaningful mathematics learning within his or her classroom.  I am not saying that every teacher should have the same definition of meaningful learning that I do, what I am saying is that every teacher should have a definition of meaningful learning and should then use that definition to guide lesson planning.  (Note:  I believe this definition must stem from a combination of personal experiences and values and the study of relevant literature and research.)  No matter how a teacher then defines meaningful learning, if the teacher is clear and uses this personal definition as a guide for planning, the lesson will be more effective.  I believe that the use of this model will guide teachers at any level of experience through a process that will result in more effective, meaningful lessons and more active, motivated students.  The following is a more detailed discussion of each of the components of the model.

Outside Influences
	I would define outside influences to the classroom as those influences that the teacher traditionally has little control over.  I would hope to see that change in the future, but for now, most individual teachers do not have much input into curricular policies, professional research, systemic expectations, and community values.  Certainly, all of these aspects of education affect, either directly of indirectly, an individual teacher's classroom.  I think it is important for a teacher to be aware of the strength of the impact these factors are having on his or her classroom, in what manner are they affecting the classroom, and how he or she can get involved with the future direction and impact of these factors.

Teacher Knowledge and Beliefs
	In the field of mathematics education research, there has been growing popularity  of studies that examine teachers beliefs and knowledge (Thompson, 1992).  Researchers have studied how knowledge of mathematics and pedagogy affects instructional practices.  They have done the similar research in the area of  beliefs.  According to Thompson, the research done in the area of knowledge has been much more substantial than in the area of beliefs.  While the results of this research are relevant and interesting for classroom teachers, I do not think it takes a brain surgeon to understand that what a teacher knows and believes about mathematics, about learning, and about teaching,  will affect the manner in which he or she plans for and conducts lessons.  Kuhs and Ball (1986) offer an four dominant and distinctive view of how mathematics should be taught:
1.  Learner-focused:   mathematics teaching that focuses on the learner's personal construction of mathematical knowledge;
2.  Content-focused with an emphasis on conceptual understanding:  mathematics teaching that is driven by the content itself but emphasizes conceptual understanding;
3.  Content-focused with an emphasis on performance:   mathematics teaching that emphasizes student performance and mastery of mathematical rules and procedures; and
4.  Classroom-focused:  mathematics teaching based on knowledge about effective classrooms. (p.2)
These four views are certainly not the only ones to consider, but they do demonstrate how beliefs about math, learning, and teaching, all work together to form a style of classroom instruction.  I am hopeful that the underlying definition of meaning can also be seen in these four examples. 

Student Knowledge and Beliefs
	What Kuhs and Ball fail to address is student knowledge, student perceptions, and teachers' perceptions of the students.  McLeod & Ortega(1993) state "unfortunately, there is very little research on secondary teachers' knowledge of students and students' mathematical learning that relates teachers' understandings of their students' learning theories or cognitive psychology to their actions in the classroom and student achievement" (p. 252).  They go on to state "unless teachers are knowledgeable enough about their students to use that information when deciding on classroom actions, even the best-intended moves may not result in desired reactions."  This is my point exactly.  even the best intended lessons will not be as effective as expected if student knowledge, beliefs, and attitudes are not taken into account.  Teachers must take time to construct a model of their students and then apply this model in the lesson planning process.

Components of the Lesson
	In planning a meaningful lesson, there are many elements to consider:  content, long term goals, connections, pedagogy, resources, assessment, and affective issues.  In my opinion, planning for meaningful instruction must incorporate all of the above components.  
	The choice of content (what topic to teach?) may seem like one that the teacher has little control over; however, I perceive this decision as very teacher controlled.  A teacher under the guise that the county or state guidelines have predetermined what her or she teaches is not being realistic or is not holding him or herself accountable for curriculum choices in the classroom.  For example, a teacher may be told that teaching slope is a mandatory objective.  Well, that means the teacher can present slope strictly in a Cartesian plane, as an algebraic manipulation, as a way of describing rates of change, etc.  For me, these are content choices and they are choices that teachers make on a daily basis.  These decisions should stem from an introspective look at what a teacher knows and values mathematically and what he or she feels is important for students to learn meaningfully.
	Having long term goals or a unifying themes (i.e. function, modeling, problem solving, applications, etc.) is an important aspect of classroom instruction.  When a new lesson is planned, these long terms goals should constantly be examined and evaluated.  There is a tendency to make long term goals at the beginning of a school year and not refer to them again.  I argue that for instruction to be meaningful, it must be in line with the teacher's goals.  In keeping instruction in line with these long term goals, connections are made more frequently and more easily.  A teacher who strives to connect mathematical topics either to previously studied mathematics or to make interdisciplinary connections provides students with opportunities to learn more meaningfully.
	Other important components of a lesson to make decisions about are the pedagogical methods that will be employed and the tools and resources that will be used to promote meaningful learning.  Rene Thom (1973) stated "all  mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics" (cited in Thompson, 1992, p. 127).    I would add that it also rests on a philosophy of learning and teaching.  If a teacher has formed a personal definition of meaningful mathematics learning, the pedagogical decisions should be clearer and easier to make.  How to best teach mathematics meaningfully  should stem from one's beliefs about what constitutes meaningful mathematics. 
	According to the NCTM Assessment Standards (1995), assessment should be ongoing, equitable, a part of the lesson, be matched closely the classroom objectives, and be reflective of the types of mathematics students are engaged in.  A simple interpretation of this would be:  Do not use multiple choice teats at the end of a unit if you have investigating and analyzing data through open-ended discussions and presentations.  Assessment needs to be aligned with decisions about content and pedagogy.  Teachers need to reflect on assessment techniques while they are planning the lesson.  Effective and informative assessment cannot be an after thought.  The authors of the El Salvadorian seventh grade program of study (1996) offer the following analogy as they promote formative (throughout the learning process) evaluation:
When the summative (at the end of a unit) evaluation is used...it is like an autopsy performed to determine the cause of death.  It could be used to better understand mortal sickness and to save the life of future patients, but it is not much help to the actual patient.
Assessment decisions should be in line with an individuals personal definition of meaningful mathematics learning.  A teacher needs to be assessing what he or she values and desires students to learn.  Assessment sends a message to students about what the teacher values as meaningful; therefore, considerable thought and time should go into the development and incorporation of assessment into a mathematics lesson. 
	There are also affective issues to consider when planning for meaningful instruction.  The models that teachers have developed of their students should help  facilitate lesson planning that accounts for and incorporates affective issues of meaningful mathematics learning.  Ortega (1993) claims "our dual challenge as teachers is to prepare students to expect some frustration and anxiety during problem solving and to learn how to deal with these feelings as the expert problem solvers do" (p. 33).  Not only are affective issues such as anxiety and frustration ones to be considered, but so are cultural and gender issues, self-esteem, attributions, beliefs about mathematics, beliefs about learning mathematics, and motivation.  These issues cannot be ignored if meaningful instruction is to occur.  
	In summary, there are many components of planning for meaningful mathematics instruction to be considered by individual teachers.  Teachers should be encouraged to use as resources research, peer discussions, students' perspectives, and their own personal beliefs when planning for instruction.  All the components discussed above are essential to meaningful learning and all need to be in line with an individual core belief about meaningful learning.  Assessment needs to match content, content needs to match long-term goals, goals need to match pedagogy, pedagogy needs to match resources, etc.  and all of these components need to be in line with what teachers know about themselves and their students.  The model I developed for instructional planning should guide teachers through a process of lining up beliefs and actions that will result in meaningful instruction.

Implications for Teacher Education
	If you read the model from right to left, you will find the process I believe most educators go through as they develop professionally.  I think it is the "expert teachers" who have reached the stage of development where they can define their own beliefs about learning, mathematics, etc.  It does not have to be this way.  These definitions can be formed earlier and the process of continually reflecting on and/or redefining  beliefs can begin in pre-service education.  Beginning and pre-service teachers should be challenged with the same questions.  They should study and discuss ideas of learning, meaning,, students perceptions, etc. before they begin planning lessons and discussing pedagogical decisions.  I know that the argument would be that those discussions mean little to someone without experience.  I argue that once an initial belief about meaningful learning is developed, pre-service and beginning teachers will have a framework with which to interpret new information about instructional planning.  I believe that the processes of developing and personal meaning of meaning, a theory about learning, a philosophy of education, and the realization that these definitions and beliefs are dynamic and will continue to develop and grow with time and experience,  is crucial to the development of professional mathematics educators and to the reality of meaningful mathematics lessons in classrooms.

Personal note:  During my  two and a half years of pre-service training in mathematics education, six years of teaching, and three years of master's degree work in mathematics education,  I have never been as challenged to define what I believe about mathematics education as I have been this first year of doctoral study.  Writing this exam gave me the opportunity to synthesize all that I have studied and discussed about learning, about mathematics, about teaching,  about myself, and about the field of mathematics education.  I am looking forward to returning to the classroom in August.  I feel that I have so much more to offer as a mathematics educator.   

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