teachers conceptualizing student achievment

8/10/2019 Teachers Conceptualizing Student Achievment

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Teachers Conceptualizing

Student AchievementRobert E. Stake

Published online: 25 Aug 2010.

To cite this article:Robert E. Stake (2002) Teachers Conceptualizing Student

Achievement, Teachers and Teaching: theory and practice, 8:3, 303-312

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Teachers and Teaching: theory and practice, Vol. 8, No. 3/4, 2002

Teachers Conceptualizing Student

AchievementROBERTE. STAKE

ABSTRACT Although teachers vary widely in their perceptions of education, studentachievement, and methods of assessment, more than psychometricians and other people

they concentrate on student performance of tasks. To the teacher, grades are an indicationof successful acquisition of particular knowledge and an increase in skill. Achievement isnot thought of so much as enhancing a trait or increasing an ability, but as successfullycompleting the task. They visualize the experience, more so than the competence. Whenemphasis is given to standardizing curricula and testing, even though goals and standardscan be expressed as task performance, the technology of testing and school reform devisesindicators of success in terms of human ability. Ability is generalized; task performance isparticularized, contextualized. When the success of teaching or schooling is interpreted interms of test scores, the teacher is pressed to reconceptualize teaching and, directly or

indirectly, to teach for the test.

Introduction

Michael Huberman and his family lived in Cambridge, MA, USA, in 199195.With a group of some 30 very special people, he and I worked on the Americanside of an Organization for Economic Co-operation and Development (Paris) and

National Science Foundation (Washington, DC, USA) project identifying success-ful national science and mathematics education innovations around the world andpreparing case study reports of them. Under the direction of Senta Raizen, oureight cases were published in 1997 under the title Bold Ventures (Raizen & Britton,1997). Michael was the technical consultant for case study methods but alsoworked with Sally Middlebrooks and James Karlan to prepare case studies on theVoyage of the Mimi and the Kids Network. I worked with Douglas McLeodstudying the development of education standards by the National Council ofTeachers of Mathematics (NCTM). It was a ne collection of chapters, subse-

quently unnoticed, needing perhaps something like the $400,000 the Council paida public relations rm to publicize its standards.

Michaels interest in the circumstances of teaching was deep and abiding. Hewas particularly attentive to efforts made by those outside the classroom tochange the way a teacher teaches, including the National Science Foundation. Oneof the circumstances he and I pondered was the sense in which teachers concep-

ISSN 1354-0602 (print)/ISSN 1470-1278 (online)/02/03/040303-10 2002 Taylor & Francis Ltd

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304 R. E. Stake

tualized standards such as the NCTM Standards, and what they consideredevidence that standards were being met. The conversation at the time took placeagainst the background of innovations in mathematics and science instruction thatI already mentioned. In the present article, I will focus on the teaching ofmathematics, although similar points could be made about other school subjectmatters. I take this opportunity to record my recollection of the meat of ourconversations about teacher perceptions, hoping that Michaels inspiration helpsme to incorporate his views.

The Teachers Conceptualization of Student Achievement

Experienced teachers have a particular way of looking at education, drawingvigorously on their own behavior in the classroom (Doyle, 1979; Shavelson &

Stern, 1981). For example, they may think of students knowledge of trigonometryin terms of the tasks they assign. They conceptualize knowledge and skill as theknowledge and skill they teach. To be educated in mathematics, in this view, is tohave completed years of assignments and done well on the tasks included inexaminations.

I speak too generally. Teachers are not of one mind. They conceptualizeeducation in different ways. Some think more like epistemologists or curriculumdesigners or psychometricians, but most of them, according to my eldwork,think of education in terms of scholastic production. While they are well aware

that students learn much outside of school, these teachers tend to think the degreeto which they are educated is the degree to which they have succeeded in schooland college and further training.

Most experienced teachers have a far-reaching conceptualization, developedover time, of the mix of ideas and behaviors that constitute a course. Thatconceptualization is based on the teachers experience with such factors as:

The calendar and time allotments. What topics should be covered.

The relationship and interdependence of topics. The nuances and sub-classications of the topical eld. Diverse applications of topics. The relevance of topics to standardized testing. Opportunities for enrichment and cooperative learning. Nurturing independent thinking and self-directed learning. Ways of increasing motivation and decreasing discouragement. What will be the stumbling blocks.

How socialization and conict pre-empt academics. What experience and vitality the students will bring. The expectations of students and parents and other teachers, and more.

Most teachers are far more able to teach sensitively in this complexity than theyare to describe it. To be an analyst of pedagogy is neither required nor evidenceof teaching competence. Drawing on a quarter century of classroom research




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Teachers Conceptualizing Student Achievement 305

(Stake & Easley, 1979; Stake et al., 1993, 1997), I speak specically of the com-plexity of the teachers own perception of the teaching of mathematics. Some referto this perception as `psychological perception (Bruner, 1962) as opposed to`sociological perception (Schrag, 1992) and to `logical structure (Ausubel, 1963).Considerably more complex than psychological perception, the logical structure ofmathematical knowledge (Romberg, 1992) is the collective perception of math-ematics education by experts in the NCTM and elsewhere.

Somewhere in the mind of each mathematics teacher is an evolving inventoryof topics to teach. Each topic alone can be as intricate as a tree, with bigger andsmaller branches, twigs and lacy buds and leaves, individually dispensable butcollectively vitalizing the tree. The parts are open to personal interpretation. Somewriters would use trunk and limbs to represent classications of goals, objectives,chapters, and types of problems as structure for the emergent learning of math-

ematics. Others would use trunk and limbs to represent the relationship amongideas. Still others would emphasize the connection of mathematics to experiencein academic disciplines and to preparation for such elds of work as accountingand engineering. Whatever the theoretical representation, it is in actual teachingthat the teacher represents the categories, relationships and connections. Morethan anything else, it is the teachers comprehension of subject matter as `mani-fest-in-action, content embedded in process, that distinguishes a teacher view ofmathematics achievement.

The inventory of mathematics to be taught is the practical epistemology of

education. Comprehensiveness and integrity of subject matter and topical evol-ution are not issues for most teachers. They are not reading John Dewey or JeanPiaget or John von Neumann. Few mathematics teachers have a philosopher ormathematician to chat with. For them, the authority of mathematics lies in syllabi,textbooks, worksheets and testseach with its leaning toward simplication, andeach increasingly bent on raising achievement scores. But it is not all teach to thetest. Within a still vital autonomy existing in most classrooms, the complexity ofteaching survives. It draws from an ideology of mathematics education rmly

embedded in the minds of the teachers (Lundgren, 1979; Darling-Hammond, 1990;Romberg, 1992).

The Establishment View of Student Achievement

The teachers view of mathematics education is not the ofcial, establishmentarianview. Ofcial documents setting forth the purpose of schools focus on studentcompetencies. The competencies are dened in the words of the psychologists of

three decades back who touted behavioral objectives. In the US today, statedepartments of education and school districts continue to reify documents identi-fying such ofcially approved academic goals. The Georgia Board of Education,for example, adopted student goal statements that identify the skills and attitudesthat a graduate of Georgias educational system should have achieved in school.State board members testied that instructional programs in the public schools




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306 R. E. Stake

should provide each individual with opportunities to develop abilities so thathe/she:

communicates effectively uses essential mathematics skills

recognizes the need for lifelong learning has the background to begin career pursuits participates as a citizen in our democratic system, etc.

When I studied it, the mathematics section of the Georgias Quality Core Curricu-lum consisted of `objectives relating to concepts, process skills and problemsolving at each grade level, kindergarten through eighth. In Grades 912, objec-tives are given for each mathematics course. Mathematics began and continues tobe a way of organizing ones world, through the study of quantity and space, their

properties and the relationship[s] within and between these concepts. Mathemat-ics is rst experienced as a language created to describe the world, accompaniedby rules that govern its use. In Georgia, Algebra I was to explore Topics/Concepts such as:

E. Polynomials13. Identies polynomial expressions.14. Adds and subtracts polynomials.15. Uses of laws of exponents necessary to perform polynomial operations, etc.

In this way, ofcial curricular statements identify content and behaviors, repre-senting mathematics education in language that is more rule-based, that are moregeneral and abstract than does the teacher.

Conceptualization of Student Achievement by the Psychometrician [1]

Experienced test makers have a particular way of looking at education, throughtheir conceptualization of traits and abilities. Knowledge of trigonometry, for

example, they see in terms of the abilities of students. The purpose of the test isto measure the ability of a student to solve trigonometric tasks. These are more orless the same tasks the teacher thinks of when conceptualizing the nature of aneducation, but the psychometrician sees them less as representing a sector of thedomain of mathematics teaching and more as a sampling of students scholasticability. For the teacher, each subtopic or task is an important acquisition, notinterchangeable with others. A good sampling of topics gives face validity andmakes consumers happy. But for the test maker, mathematics achievement is ahomogeneous construct. If half the domain called Trigonometry were transferred

to Vector Science, the psychometrician would not presume that a different testnecessarily would be needed.

I speak too generally. Like teachers, test makers are not of one mind. Theyconceptualize education in different ways. Some think more like epistemologistsor curriculum designers or teachers, but many of them think of education in termsof scholastic aptitudes. They are well aware that students learn diversied content




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1. 203 1.81 325 ?

2. (1/5)(20)(9)1 325 ?

3. 23 203 9

101 325 ?

4. y5 1.8x1 32. Solve for y if x5 20.

5. Convert 20C to Fahrenheit. F5 (9/5)C1 32.

6. Convert 20C to Fahrenheit. C5 (5/9)(F232).

7. Ann wants to know todays temperature on the Fahrenheit scale. Her thermometer reades 20degrees Celsius. What is the Fahrenheit temperature?

FIG. 1. A family of seven mathematical tasks.

in school, but test makers tend to think that the degree to which a person iseducated is the degree to which he/she has maximized a potential for furtherlearning.

Seven mathematics tasks are shown in Fig. 1. To a test maker, these tasks arepoints on a single scale; differing only in difculty. To a teacher, each of the tasksrequires separate knowledge. Across a large group of students, the statisticalcorrelation among the seven would run high but each task requires its own

understanding of operations. The mathematics teacher provides instruction oneach task. Getting any six of the items right does not assure getting the seventhright. To a teacher, mathematics achievement is not just getting the best score onthe test; it is understanding and performing the work.

These seven tasks cut across several content domains (Hively et al., 1973), yet ateacher might include all of them within a single lesson, within a single objective,or refer the solution of each to a single page of textbook. Each task is unique, aspecial variation on the others. Each will be more or less well understood bystudents and thus more or less difcult. Although different in form, they belong

to a family. The family is not dened by mathematical operations as much as bythe practical problem of dealing with two temperature scales, Celsius and Fahren-heit. To a teacher, all seven need teaching and testing. To a test specialist, any oneof them is mathematical enough.

Representing Student Achievement.

Like all people, teachers use simple representations. Their course outlines and

lesson plans briey list topics and activities. To satisfy the requirements ofadministrators or to talk to visiting parents, they sometimes refer to lists ofobjectives such as those for the State of Georgia. But in thinking about how andwhat they will be teaching, teachers work at a much higher level of specicity andcomplexity.

Within mathematics education, as within all of education, there is far more




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FIG. 2. A map of mathematics education content.

interweaving and interdependence of meaning than is apparent in a test score oron a list or on a `content by behavior grid (Wilson, 1971, p. 646). What if we tried

to represent the similarity of mathematics topics as a function of distance, whereclose together means similar? The map in Fig. 2 is one such representation. Itcould refer to the knowledge of a single student, the achievement of a group ofstudents, or the logical structure of the eld. Here, the topics of trigonometry arecloser to geometry than to arithmetic. If we had more detail, we would expect tosee `percentages lying closer to `fractions than to `probability. Many equallygood maps of this content could be presented. A two-dimensional map raisesmany good questions but, for providing insight into the relationships of math-

ematical domains, turns out to be almost as unsatisfying as a list. The relation-ships overwhelm the mapping.When we analyze what a teacher is doing, we nd topics and activities

connected in logical ways as if all were mapped there in the teachers mind. Whenwe ask for it, the teacher cannot produce the map beyond certain points thatindicate what teaching ts where. Indirectly more than directly, the teacher hastransformed complex epistemological relationships into course schedule withon-the-spot responsiveness. When we analyze the thrust, we nd the teaching isnot aimed at developing some general mathematical ability, but at developing

specic topics and skills for solving specic kinds of problems. The inventory isthe tacit map by which the pursuit of knowledge is rationalized.

Mathematics teachers require students to perform tasks. They allocate greatblocks of time to operations and exercise work. Their conceptualization ofmathematics teaching is process oriented more than outcomes oriented. Theteachers strive for high-quality experience, immersion in the topic, honing of the




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FIG. 3. An impressionistic representation of a teachers approach to teaching the PythagoreanTheorem.

particular solution. Few mathematics teachers start with thinking about how tomake children `numerate or (until harassed) how to get better scores on anachievement test. Their rst aim is to help children gain command of a far-reach-ing, little-told inventory of subject matter, outlined perhaps as the NCTM pro-posed but extending to a network of detail, which itself as salient as the majorclassications.

How does a teacher of 11 year olds approach the lesson? Figure 3 is my

impressionistic representation of choices made by a teacher as to what to teachtomorrow about the Pythagorean Theorem [2]. The topic is identied in the stateslist of learner objectives, in the district curriculum guide and in the textbook theteacher is using. To a degree, the textbook author denes what will be taught but,especially in recitation (Shavelson & Stern, 1981; Darling-Hammond, 1990), theteacher modies course content to t the situation, noting especially the frame of




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310 R. E. Stake

mind of the students. Reecting on the many topics pertinent to the course(Topography A, bottom of Fig. 3), the teacher considers the acts of teaching interms of facts, concepts, relationships, and applications of the Pythagorean The-orem. Distribution B represents a closer look at what is most relevant for theseparticular students (of course, it will be different for each child). The teacherdraws elements from several knowledge bases (Circles C, D, and E) to obtain asmall selection to teach (Plate F) and then, reecting on previous exchanges inclass, thinks about learning difculty (Cylinder G) and imagines herself/himselfserving up the content (Tray H) for this class. The teacher anticipates a smallpresentation with graphics, reading, seatwork and homework. When this occurs,ideas are modied while the conversations of instruction go on. The process isshaped, of course, by the teachers overall conceptualization of mathematicseducation (Lampert, 1988; Easley & Easley, 1992).

Figures 2 and 3 are simplistic, but not really more so than of my earlier lists ofclassroom conditions and state goals. Graphic technology to represent pedagogyand epistemology is not highly developed [3]. Classication systems and content-skill grids are common in district-level curriculum ofces but there are fewdevices to represent conceptual links between topics and potentially to guidepedagogical moves from one content to another. Yet, just as ancient travelersreached destinations before there were maps, teachers teach without them. Intu-itively, good teachers merge topical paths, capitalize on personal experience, anddraw out and preserve the youngsters line of thought.

The differences among conceptualization of student achievement by teachers,authorities, and test makers provide insight into current difculties of usingstandardized testing for school reform. A teacher does think about a panoply oftasks and a panoply of student competencies, but translating one into the other isneither easy nor common. Teachers pay little attention to performance on individ-ual test items (tasks) because, lacking diagnostic validity, such tasks give littleguidance to instruction. They were chosen because they discriminated amongstudents with high and low aptitudes, not because they indicate the most critical

learning content. Teachers see the authorities pressing for recognition of theirmatrices of competency because the competencies are more closely linked to thetested aptitudes [4], but the teachers have an enormous investment in conceptual-izing the tasks to be taught, some of which must be cut if the teacher maximizesadherence to the syllabus and readiness for the tests.

Recalling conversations with Michael, I have described the teaching and learn-ing of mathematics as intricately detailed. Other subjects have different conceptualstructures, no less intricate. From their words and activities in the classroom andin long unstructured interviews, I have found that teachers conceptualization of

student achievement greatly inuences their planning, instructional strategy, andassessments. Although the writings of subject matter specialists, school districtsyllabi, textbooks and tests can be claimed to be built on more powerful epistemo-logical structures, these formal conceptualizations of education fail to identifymany characteristics of achievement important to teachers for whom it is import-ant to visualize the tasks taught and learned.




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Notes

[1] Testing, broadly considered, is the presentation of certain tasks with responses judged asright and wrong or above and below criterion. Both teacher-made and psychometricachievement testing extrude student performances to be interpreted in summary as aboveor below those of other students. The standardized test score indicates an absolute score

on a pedagogically uninteresting selection of tasks, but a powerful indicator of thestudents relative standing among other examinees. The test is not intended to bediagnostic. The score is taken as an index of achievement, a datum, a bit of information.For most people, testing is seen mainly as information gathering, but the limits of thatinformation are not widely known (Stake, 1995).

[2] Figure 3 is not a research nding, it is merely a euphemism, a view representing whatexperienced teachers have appeared to do in my observation. When asked, they seldomclaim to be involved in such detailed analysis. And yet such operations can be observed.The point again is that the intuitive working of teaching is highly complex with fargreater texture than the goals stated by the state or district.

[3] Some of the best works to date are Driver (1973), Gowin (1990), Jonassen (1982), Sato(1991) and School Mathematics Study Group (1961). These works analyze either instruc-tion, epistemology or cognitive development; they do not adapt nicely to the `conversa-tional exchanges of classrooms.

[4] Many of the tests are labeled as achievement tests and they have a patina of curricularlanguage, but they attain a high discriminatio n quotient by being good indicators ofscholastic aptitude. They indicate who is good at taking tests and they predict who willbe good at taking tests in the future (Stake, 2001).

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