ERIC ED460828: NCRMSE Research Review, 1991-1994.

119 Pages·1991·2.6 MB·English

by  ERIC

DOCUMENT RESUME
ED 460 828
SE 061 819
TITLE
NCRMSE Research Review, 1991-1994.
INSTITUTION
National Center for Research in Mathematical Sciences
Education, Madison, WI.
SPONS AGENCY
Office of Educational Research and Improvement (ED),
Washington, DC.
PUB DATE
1991-00-00
NOTE
118p.; Published three times a year.
AVAILABLE FROM
National Center for Research in Mathematical Sciences
Education, 1025 W. Johnson Street, Madison, WI 53706. Tel:
608-263-7582; Fax: [phone].
PUB TYPE
Collected Works
Serials (022)
JOURNAL CIT
NCRMSE Research Reviews; n1-3 Oct 1991-Fall 1994
EDRS. PRICE
MF01/PC05 Plus Postage.
DESCRIPTORS
*Educational Assessment; Educational Change; Elementary
Secondary Education; Mathematics Curriculum; *Mathematics
Education; Middle Schools
IDENTIFIERS
National Center for Research in Math Sciences Educ
ABSTRACT
This document consists of the first three years worth (9
issues) of the serial, NCRMSE Research Review: The Teaching and Learning of
Mathematics. Each issue reviews the research of the National Center for
Research in Mathematical Sciences Education (NCRMSE). Major articles
appearing during this period (1991-1994) are as follows: "Perspectives
on
Assessment"; "A Study of Reform of Mathematics Education"; "Cognitively
Guided Instruction"; "Reflective Practitioners Reform School Mathematics";
"The Teachers and Algebra Project"; "New Book Considers Domain of Rational
Numbers"; "School-Welfare Reform and Teaching/Learning of Mathematics"; "What
Gets Graded Is What Gets Valued"; "Geometry in the Primary Grades";
"Integrating Statistics into the School Curriculum"; and "Gender and
Mathematics Education Research." (ASK)
Reproductions supplied by EDRS are the best that can be made
from the original document.
NCRMSE Research Review:
The Teaching and Learning of Mathematics
Volumes 1-3
(Oct. 1991
Fall 1994)
U.S. DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
EDUCATIONAL RESOURCES INFORMATION
CENTER (ERIC)
O This document has been reproduced as
received from the person or organization
originating it.
O Minor changes have been made to
improve reproduction quality.
Points of view or opinions stated in this
document do not necessarily represent
official OERI position or policy.
2
BESICOPYAVAILABLE
NCRMSE RESEARCH REVIEW
The Teaching and Learning of Mathematics
October 1991
Volume 1, Number 1
NCRMSE Begins Five-Year Program
In This Issue:
on the Implementation of Re-
You are reading the first issue of
group
New Book Cites
form; Judith Sowder of San Diego
the NCRMSE Research Review: The
Significant Advances
State University directs the group
Teaching and Learning of Mathemat-
on the Learning / Teaching of
ics. Designed as a newsletter, it will
in the Study of Teaching
Quantities; Richard Lehrer of the
inform the mathematics education
2
and Learning
University of WisconsinMadison di-
research community, educational
rects the group on the Learning /
policy and decisionmakers, and edu-
Perspectives
Teaching of Geometry; and Tho-
cators about the research and re-
3
mas A. Romberg of the University
lated activities undertaken by the
on Assessment
of WisconsinMadison directs the
the National Center for Research in
group on Models of Authentic As-
Mathematical Sciences Education.
Reviews of
sessment. A Statistics group will be-
The Research Review will highlight
4
NCRMSE Research
gin its work next year.
specific NCRMSE programs, pro-
vide summaries of the knowledge
Each working group involves a
A Framework for
developed by the programs, and
small group of productive re-
bring recent national and interna-
Authentic Assessment
searchers who share the same field
tional developments in mathematics
6
in Mathematics
of study, set priorities for research,
education to the attention of read-
recruit and train students, commu-
ers.
nicate with one another, and moni-
A Study on the
tor the rapidly changing structure
All NCRMSE research programs
Reform of Mathematics
emphasize building unified para-
in a field.
Education
11
digms of study. Research on class-
Two of the working groups-Imple-
room instruction, student learning,
mentation of Reform and Models of
curriculum, and assessment, accord-
Authentic Assessment-cut across or
ing to NCRMSE Associate Director
interrelate with the five content
NCRMSE Research Review:
Thomas P. Carpenter, often have
working groups. Each content-fo-
The Teaching and Learning of
been conducted in ways that isolate
cused working group has devel-
Mathematics is published by the Na-
the disciplines, creating separate
oped preliminary research goals.
tional Center for Research in
and distinct streams of inquiry. The
Mathematical Sciences Education,
NCRMSE plan of research insures
While the two cross-cutting working
Wisconsin Center for Education Re-
that teaching, learning, assessment,
groups have designed preliminary
search, University of Wisconsin-
and curricular reform are ap-
research goals, additional tasks will
Madison. The Center is funded
proached from an integrated per-
develop though their relationships
primarily by the Office of Educa-
spective.
with the content-specific working
tional Research and Improvement,
groups. The Implementation of Re-
Seven working groups carry out the
U. S. Department of Education
form Working group will examine
NCRMSE activities: Thomas P. Car-
(0ERVED). The opinions ex-
how educational reform becomes in-
penter and Elizabeth Fennema,
pressed in this publication do not
tegrated into classroom practice. It
University of WisconsinMadison, di-
necessarily reflect the position or
is undertaking specific studies of ef-
rect a group on the Learning /
policy of OERI/ED; no endorse-
forts to alter curriculum and prac-
Teaching of Whole Numbers;
ment of OERI or the Department
tice in mathematics classes. These
James Kaput of Southeastern Massa-
of Education should be inferred.
studies will identify the kinds of ex-
chusetts University directs the
Send correspondence to NCRMSE,
periences, resources, and support
group on the Learning / Teaching
1025 W. Johnson Street, Madison,
systems teachers need if they are to
of Algebra and Quantitative Analy-
WI 53706. Telephone (608) 263-
carry out the reforms called for in
sis; Walter Secada of the University
7582; FAX [phone].
the NCTM Standards.
of WisconsinMadison directs the
National Center for Research in Mathematical Sciences Education
The Models of Authentic Assess-
ment Working Group is identifying
New Book Cites Significant
a variety of models of assessment
practices that are aligned, or in
Advances in the Study of Teaching
agreement, with the reform goals
set out by the NCTM Standards. It
and Learning*
will develop procedures for rating
the validity, reliability, and utility of
"Connecting Mathematical Teaching and Learning," one of the chap-
the models. The procedures will be
ters in Integrating Research on Teaching and Learning Mathematics,
used to judge models designed for
presents initial discussions on the development of a unified para-
program evaluation by states and
digm for the study of the teaching of mathematics that incorporates
schools or for instructional deci-
both cognitive and instructional research.
sions by teachers. After potentially
During the last decade, significant advances were made in the study
useful procedures are identified,
of student learning and problem solving in mathematics, as well as
the group will construct or adapt
the study of classroom instruction. Mathematics educators have been
the assessment models, examine
concerned that these two research efforts have been conducted as
them for sensitivity to cultural and
there
separate fields of inquiry. A number of them have agreed that
linguistic diversity, prepare aggre-
is an increasing need for an integrated research program that unites
gation and reporting procedures,
the two areas. Each of the book's eight chapters presents the perspec-
and demonstrate the viable features
tive of its author on integrated research programs. The chapters in-
of the assessment procedures to
clude:
educators.
"Research and Cognitively Guided Instruction"
by Thomas P. Carpenter and Elizabeth Fennema;
Thomas A. Romberg is responsible
"Diversity, Equity, and Cognitivist Research" by Walter G. Secada;
for the overall direction of the
Research on Learning and Instruction in Mathematics:
NCRMSE. Joan Daniels Pedro is as-
The Role of Affect" by Douglas B. McLeod;
sistant to the NCRMSE director,
"Curriculum and Teacher Development: Psychological and
and Donald Chambers is director of
Anthropological Perspectives" by Paul Cobb, Erna Yackel, and Terry Wood;
dissemination. A National Advisory
"Connecting Mathematical Teaching and Learning" by Magdalene Lampert;
Panel of seven members advises the
"Methodologies for Studying Learning to Inform Teaching"
NCRMSE on the management of its
by James Hiebert and Diana Wearne;
research programs and reviews its
"Intermediate Teachers' Knowledge of Rational Number Concepts"
by Thomas R. Post, Guershon Harel, Merlyn J. Behr, and Richard Lesh; and
work. Members of the Advisory
"Improving Research in Mathematics Classroom Instruction"
Panel include Merlin Wittrock,
by Douglas A. Grouws.
chair, University of California-Los
Editors of the book are Elizabeth Fennema and Thomas P. Carpen-
Angeles; Robert Davis, Rutgers Uni-
of Wis-
ter, professors in the School of Education at the University
versity; Audrey Jackson, Parkway
consin-Madison, and Susan J. Lamon, an assistant professor in the
School District, St. Louis; Harvey
Department of Mathematics, Statistics, and Computer Science at Mar-
Keynes, University of Minnesota;
Edu-
quette University. It is part of a series, Reform in Mathematics
Jeremy Kilpatrick, University of
cation, edited by Judith Sowder of San Diego State University.
Georgia; Mary Lindquist, Colum-
bus College, Columbus, Georgia;
Integrating Research on Teaching and Learning Mathematics
and Edward Silver, University of
Elizabeth Fennema, Thomas P. Carpenter and Susan J. Lamon, Editors
Pittsburgh.
1991, 142 pages, $12.95 paperback, $39.50 hardcover
Available from: State University of New York Press, c/o CUP Services, P.O.
The NCRMSE is funded by grants
Box 6525, Ithaca, NY 14851; phone [phone].
from the Office of Educational Re-
*The book was developed at the National Center for Research in
search and Improvement, United
Mathematical Sciences Education with funding from the Office of
States Office of Education, Washing-
Educational Research and Improvement, U.S. Department of Educa-
ton, D.C. In addition to publishing
tion (0ERVED).
the NCRMSE Research Review and
carrying out its 5-year research pro-
gram, NCRMSE distributes re-
search reports, publishes
cation, University of Wisconsin-
For further information on
monographs relating to mathemat-
Madison, 1025 W. Johnson Street,
NCRMSE contact Donald Cham-
ics education, and provides infor-
Madison, Wisconsin, 53706, or call
bers, Director of Dissemination,
,
mational programs to the
[phone].
NCRMSE, Wisconsin Center for
mathematics education community.
Education Research, School of Edu-
4
National Center for Research in Mathematical Sciences Education
2
the quality of educational systems'
Perspectives on
velopment of national examina-
performance. Those responsible for
tions, "American Achievement
Assessment
the revision of the test are weighing
Tests," that will be used to assess stu-
the questions of how well it pres-
dent performance and to indicate
ently is aligned with the NCTM
the effectiveness of educational sys-
National and international attention
Standards and how well it addresses
tems. The concern generated by
increasingly is focused on the role of
the concerns regarding the limita-
this requirement led to the creation
assessment in education. The test
tions of multiple choice items.
of a National Testing Council in
scores of American students, particu-
July, 1991. The council consists of
larly the mathematics scores they
The New Standards Project in-
32 members, 22 of them to be ap-
achieve on standardized achieve-
tends to create a national examina-
pointed by U.S. Secretary of Educa-
ment tests, once were widely used to
tion system, as compared with a
tion Lamar Alexander.
compare individuals, schools, and
single examination. It will begin by
states. In the past, these compari-
involving a large number of people
An International Commission on
sions-developed with psychometric
in establishing consensus on a
Mathematics Instruction (ICMI)
sophistication-inspired public trust
framework. The framework will
held a conference on Assessment in
and were used to support varied po-
Mathematics Education and Its Ef-
litical and educational agendas.
fects April 11-16, 1991 in Calonge,
national and international
Spain. This conference provided a
Today educational scholars in the
forum for discussion of the assess-
United States and abroad are re-
educational conferences
ment tasks and strategies used by
thinking the educational value of
have featured agendas
many countries, particularly Japan,
tests made up predominately of mul-
with a primary focus on
Russia, and Australia.
tiple-choice items or of items not re-
educational assessment
lated to the instructional programs
The Mathematical Sciences Educa-
of schools. These scholars agree on
tional Board (MSEB) held a Na-
the need to provide students, teach-
tional Summit on Mathematics
ers, and parents with information
then be used to develop an examina-
Assessment in Washington, D.C.,
about student performance in rela-
tion and standards for grading. De-
April 23-14, 1991. It was a political
tion to a set of established stand-
veloped by the Learning Research
meeting designed to acquaint policy-
ards. They also agree on the need
and Development Center and the
makers with assessment issues and
National Center on Education and
to provide administrators and pol-
had the support of such prestigious
icy makers with information about
the Economy, the project will pro-
associations as the National Acad-
how well the educational system is
duce a "first draft" of a prototypic
emy of Sciences, the National Acad-
performing. Their professional at-
Grade 4 mathematics assessment
emy of Engineering, and the
tention is directed to developing al-
framework that will have been re-
Institute of Medicine.
ternate types of assessment items,
viewed by the Mathematical Sci-
tasks, response formats, and scoring
ences Educational Board's Study
The National Center for Educa-
Group on Mathematics Assessment
tion Statistics (NCES) held an Inter-
and by the MSEB Board. By the
national Review of Preliminary
end of October 1991, the project
Recommendations for the Third In-
alternate assessment
will complete a "refined draft" of
ternational Mathematics and Sci-
strategies also will require
prototypic Grade 4 mathematics as-
ence Study (TIMSS) in May, 1991.
new psychometric
sessment materials and the needed
The meeting was planned to enable
ancillary materials such as scoring
the discussion of frameworks from
approaches
rubrics and teachers' guides.
which the questionnaires and items
for the TIMSS would be developed.
This first issue of Research Review re-
Experts from many countries partici-
rubrics. They are convinced that al-
ports on some of the NCRMSE re-
pated. The results of the meeting
ternate assessment strategies also
search on assessment. That research
may provide models of appropriate
will require new psychometric ap-
supports the need for some new
means of gathering data that will in-
proaches.
thinking about assessment-particu-
dicate how well educational systems
larly the kind exemplified by the
During the last year, national and
are performing.
New Standards Project. It also car-
international educational confer-
ries a related article by Susanne La-
The National Assessment of Edu-
ences have featured agendas with a
joie that defines authentic
cational Progress (NAEP) has been
primary focus on educational assess-
assessment, develops a framework
making state-by-state comparisions
ment:
for its identification, and provides
of levels of performance in mathe-
President Bush's education strat-
several examples of assessment
matics and other instructional areas.
egy is described in a publication,
strategies that hold promise as
The items they use have been
America 2000, released in March,
authentic forms for assessing mathe-
viewed by many as an appropriate
1991. The strategy calls for the de-
matics learning.
way of gathering data to indicate
3
Wisconsin Center for Education Research, University of Wisconsin-Madison
5
Reviews of NCRMSE Research
,07,07,e,,W,P17,,,OrMlifar ,MiffeXAMAINVEN
ing and the curriculum. When
tests were given in their school set-
Anecdotal evidence has long sug-
teachers think districts use test data
tings. If tests were given, teachers
gested that the tests required of
to make decisions about students,
were asked about their knowledge
their students by schools, districts,
teachers are much more likely to
of the basic content and style of the
or states affect the instructional prac-
consider the style and format of the
mandated test taken by their stu-
tices of teachers. Teachers, accord-
test as they plan instruction than
dents. It also asked about teachers'
ing to this evidence, "teach to the
when this is not the case.
efforts to ensure that their students
test." Several years ago the Mathe-
perform well on the tests, about ad-
matical Sciences Education Board
Fewer than 20 percent of teachers
justments they make in the curricu-
held a national conference on The
say they make no instructional
lum to focus on the knowledge and
Impact of Testing on Mathematics
changes based on test results. Those
skills on the tests, and about adjust-
Education. If tests influence what
teachers making instructional
ments in their modes of instruction
changes say that as a result of test-
in response to the style and content
ing they increase their emphasis on
of the tests. Teachers were also
testing programs can
some areas and decrease it on oth-
asked about the effects of testing on
be used to stimulate
ers. They spend additional time on
the use of problem solving and of
instructional change or
basic skills, pencil and paper compu-
calculators during instruction.
tation, topics emphasized on the
they can become
The use of mandated tests with 8th-
test, and direct whole class instruc-
impediments to change
grade mathematics students is wide-
tion. They report a decreased em-
spread. Nearly 87 percent of the
phasis on extended project work,
responding teachers indicated their
problem solving, activities involving
students were given mandated
mathematics is taught and how it is
calculators, topics not emphasized
mathematics tests. About 68 percent
taught, said conference participants,
on the test, activities involving com-
of teachers said their students took
testing programs can be used to
puters, and cooperative learning. Fi-
a district-level test while 46 percent
stimulate instructional change or
nally, those teachers who give
said their students took a state test
they can become impediments to
and another 18 percent said their
change. Three members of that
students were part of a state assess-
group, Thomas A. Romberg,
ment program. Only 13 percent of
Jeremy Kilpatrick, and Tej Pandey,
testing does have
teachers indicated their students
proposed a set of studies that would
an impact
were not required to take a man-
provide a research base and extend
on instruction
dated test in mathematics.
what is known about this issue.
Test results are used by districts
Several of the proposed studies on
and by teachers in a variety of ways.
the influence of mandated testing
The results obtained by tests man-
on school mathematics instruction
students practice test items and set
dated by districts are used, said
were recently completed by re-
aside time for students to prepare
teacher respondents, by 80 percent
searchers at The National Center
for the test are much more likely
for Research in Mathematical Sci-
than others to reduce their empha-
ences Education. One of the studies
sis on activities that involve comput-
of testing obtained information on
ers and calculators.
fewer than 20 percent of
the experiences and the percep-
tions of a nationwide sample of
teachers say they make
Testing does have an impact on in-
1,200 8th-grade mathematics teach-
no instructional changes
struction, according to this national
ers. The second study of testing ex-
survey of 8th grade mathematics
based on test results
amined the alignment of six of the
teachers. Teachers know what the
most widely used 8th-grade mathe-
tests call for and they plan accord-
matics tests with the NCTM Currku-
ingly. These results suggest that if
of districts. They use them to group
lum and Evaluation Standards for
districts adopt or prepare a test that
students by ability, compare them to
School Mathematics. The findings of
sets standards of achievement they
national norms, or assign them to
the two studies are reported here.
regard as suitable, a test with a style
special programs.
and format likely to promote the
Instructional Impact
Most teachers are provided with test
kind of mathematics the NCTM
of Testing
data for individual students at an
Standards seek, the use of that test to
item or objective level. A majority of
set standards and make decisions
The first study on the instructional
teachers think their districts are us-
about students can also foster de-
impact of testing asked mathematics
ing the test results to stress what is
sired instructional changes.
teachers whether mandated stand-
tested or to deliberately affect teach-
ardized mathematics achievement
6
National Center for Research in Mathematical Sciences Education
4
mows
process area. While 20 percent fall
To complete the analysis of the
NCTM Standards
into the Communication process,
tests, each item on each of the six
and Six Tests
only 1 percent or less fall into the
tests was categorized by content,
remaining process areas called for
process, and level using the NCTM
by the NCTM Standards, Problem
Standards for Grades 5-8. The con-
Based on the first study of testing,
Solving, Connections, Reasoning,
tent category included Numbers
we reported that tests have an im-
and Patterns and Functions. An av-
and Number Relations (nr), Num-
pact on the instructional programs
erage of 89 percent of the items are
ber Systems and Number Theory
of a majority of 8th-grade teachers.
classified as Procedural and 11 per-
(ns), Algebra (alg), Probability and
A second NCRMSE study used
cent as Conceptual.
Statistics (p&s), Geometry (geo),
standardized mathematics achieve-
and Measurement (mea). The proc-
ment tests that were identified in
If tests are to reflect the new vision
ess category included Problem Solv-
the first study. Only six of the tests
of the mathematics curriculum de-
ing (ps), Communication (com),
those identified as receiving the
veloped by Working Groups of the
Reasoning (rea), Connections (con),
most widespread district usage na-
NCTM Commission on Standards
Computation and Estimation (c&e),
tionwidewere examined.
and patterns and Functions (p&f).
for School Mathematics it will be
necessary to vary content more
And the level category included
than done at present; the processes
Conceptual (conc) and Procedural
standardized mathematics
of Problem Solving, Reasoning,
(proc).
achievement tests studied
Connections and Patterns and
There were few differences across
Functions will need to be added.
do not cover adequately the
the six tests as shown in Table 1.
An increase in the conceptual level
range of content called for
When each item was identified ac-
of their items will also be needed.
in the NCTM Standards
cording to content, process, and
And if the NCTM Standards are to
level the emphases of the tests did
be implemented by schools, schools
not vary. The percentages of the
will need to select a set of stand-
items on the individual tests fitting
ardized mathematics achievement
Our purpose was to determine
into the areas within each of the
tests different from those used by
whether the widely-used tests re-
three categories were similar to the
the majority of districts and states,
flected the recommendations of the
average percentages for all of the
or encourage the development of
NCTM Curriculum and Evaluation
tests.
more adequate tests by districts and
Standards for School Mathematics
states or by the developers of the
The results of this examination
(1989). The NCTM Standards were
standardized tests they currently
developed as a means of improving
show that the six standardized
purchase.
the quality of school mathematics.
mathematics achievement tests stud-
The Board of Directors of the Na-
ied do not cover adequately the
References
tional Council of Teachers of
range of content called for in the
Mathematics established the Com-
NCTM Standards for Grades 5-8. A
Romberg, T. A., Zarrinnia, E., & Williams,
mission on Standards for School
majority of items, 71 percent, fall
S. (1989, March). The influence of man-
into content area of Numbers and
Mathematics. The Commission
dated testing on mathematics instnIction:
Grade 8 teachers' perceptions. Madison,
drafted a document that represents
Number Relations. While 9 percent
WI: National Center for Research in
the consensus of NCTM's members
fall into the Measurement content
Mathematical Sciences Education.
area, between 3 and 6 percent fall
about the fundamental content that
Romberg, T. A., Wilson, L., & Khaketla,
into each of the remaining content
should be included in the school
'Mamphono. (1991). The alignment of six
mathematics curriculum and about
areas. In the process category, a ma-
standardized tests with the NCTM Stand-
key issues regarding student and
jority of items, 79 percent, are in
ards. Madison, WI: National Center for
program evaluation.
Research in Mathematical Sciences Edu-
the Computation and Estimation
cation.
TABLE 1
PERCENT OF NUS FOR EACH CATEGORY
PROCESS
LEVEL
CONTENT
conc
p&f proc
c&e
ma
con
com
ns
ps
geo
mea
p&s
nr
alg
84
16
0
5
0 0
91
0
4 3
7 7
SRA
82
1
10 90
0
83
0
0
6
6
4
6
6
73
11
5
CAT
sta
8
92
0
38 0 0 62
0
o
2
15
10
9
64
as
4
96
o
9
0
4
7
13
3
62
1
FIBS
1
1
88
0
12
0 79
0 0
0
5 8
15
6
21
66
MAT
85
15
2
0
71
2
0
25
CIBS
0 8 5 0
76
11
89
20
0 79
9
5
11
6
5
3
71
AVG.
1 1
1
20
2 2
0
12
18
20
11
10
7
15
6
RNG.
11
1
Wisconsin Center for Education Research, University of Wisconsin-Madison7
as well as to record them. It can be
forms of assessment. In considering
A Framework
used to assess developmental
these distinctions we must also con-
for Authentic
changes in reasoning, hypothesis
sider whether a framework for
formulations, verifications, and revi-
authentic assessment should incor-
Assessment in
sions. Technology can also serve as
porate a guide for authentic instruc-
a medium for instructional manipu-
tional activities in the classroom.
Mathematics
lation where small changes in the in-
Should we be concerned with
structional environment may
mathematical knowledge that trans-
By Susanne Lajoie
account for changes in the learner's
fers to everyday uses of mathematics
acquisition of knowledge.
or should we consider authentic
Vast differences exist between the
mathematics as something mathema-
tasks learned in school mathematics
The first four of the NCTM Stand-
ticians do in their domain?
and those mathematicians or users
ards were written as overarching
of mathematics actually carry out
goals that should be considered for
My primary focus in defining
(Pollak, 1987; Resnick, 1988; Lam-
all mathematics content at all levels.
authentic assessment in mathemat-
pert, 1990). Much of how we learn
Any specific mathematical content,
ics is to provide a robust perspective
inside the classroom is different
according to the four, should be de-
of the individual learner's under-
from how we learn outside of the
signed to provide students with op-
standing of mathematics. Several
classroom (Resnick, 1987). The fo-
portunities for mathematical
audiences are considered as I define
cus inside the typical American class-
problem solving, communication,
worthwhile mathematical tasks from
room is on what the individual
reasoning, and connections.
the mathematical educator's per-
learner can accomplish inde-
spective, followed by a description
Problem Solving
pendent of the group, or of any
of two interrelated theoretical per-
tools for learning such as calcula-
spectives on authentic activities as
Activities that give students experi-
tors. In contrast, outside-of-class-
described in the literature on situ-
ence with problem solving can
room learning situations often are
ated cognition and social construc-
emerge from problem situations.
group situations where knowledge
tivism.
These situations can be used to moti-
must be shared and where tools are
vate students and serve as a context
available to enhance or extend our
Worthwhile
in which information is learned and
knowledge. Inside the classroom stu-
knowledge is recreated across
Mathematical Tasks
dents are taught to manipulate sym-
grades. Such situations imply com-
bols and abstract principles, but
The NCTM Standards represent
plex, messy, and culturally-based
outside the classroom learning often
goals for worthwhile or essential
problems that are open to multiple
is concrete and situated in the con-
mathematics that are designed to
strategies and solutions (Zarinnia &
text in which it will be used. The
make students mathematically pow-
Romberg, in press). Problems that
term "authentic" has been used to
erful. These goals must be trans-
are messy or ill-defined can provide
suggest that some classroom activi-
lated into tasks that exemplify
more freedom for learners to pur-
ties are lacking in realism and to
authenticity. Only then can a frame-
sue questions that reflect their per-
conjure up an image of an alternate
work for authentic assessment be de-
sonal interests. Such interests can be
approach.
veloped.
promoted by providing students
with relevant, real-world applica-
Requests for more authentic class- A definition of an authentic mathe-
tions. Real-world problems often in-
room activities have led to requests
matical activity emerges from the
clude too little or too much
general assumptions of the NCTM
for authentic forms of assessment.
information; they cannot be solved
Standards. One assumption is that
These requests have come from sev-
by applying a set of routine proce-
eral audiences. They range from stu-
knowing mathematics is doing
dures.
dent, teacher, district, and state
mathematics. Doing mathematics re-
sources to a national agenda on the
fers to gathering and discovering
Problem solving with mathematics
knowledge in the course of solving
integration of instruction and assess-
involves modeling the problem and
genuine problems where knowledge
ment While the rhetoric is convinc-
formulating and verifying hypothe-
emerges from experiences that are
ing, the images of authentic
ses by collecting and interpreting
challenging but solvable. One way
activities and assessment are still im-
data using pattern analysis, graph-
precise. This article is written to
to increase such opportunities is to
ing, or computers and calculators.
stimulate discussion on ways that
provide students with experiences
Technology is a powerful tool; it
authentic assessment can be opera-
in building mathematical models,
permits learners to manipulate data
tionally defined in the area of school structures, and simulations across
and see the consequences of their
multiple disciplines. Model building
mathematics.
work in a few seconds.
and discovering mathematical pat-
The distinctions between in-school
Problem-solving activities need to in-
terns are dynamic and constructive
and out-of-school learning have im-
clude those that apply mathematics
processes. Technology can be used
plications for defining authentic
to the real world and those that
to facilitate these cognitive processes
National Center for Research in Mathematical Sciences Education
6
motes attitudes of inquiry and inves-
Reasoning
arise from the investigation of
tigation as well as sensitivity to the
mathematical ideas. Traditional cur-
Mathematics involves both inductive
inter-relationships between formal
ricula have emphasized mathemati-
and deductive reasoning. Inductive
mathematics and the real world.
cal ideas. The impetus for
reasoning is associated with mathe-
developing real and relevant prob-
Problem solving, communicating,
matical creativity or invention. De-
lems stems from the need to contex-
reasoning, and making connections
ductive reasoning involves
tualize mathematical concepts in a
can be seen as curriculum goals that
understanding the premises of a
concrete rather than abstract man-
permeate the entire mathematics
mathematical problem and reason-
ner. These real world problems may
curriculum. Specific content areas
ing logically with the given informa-
take on cultural biases depending
also need to be addressed: number
tion. Challenging problem
on the students who work with
and number relations, number sys-
situations can provide opportunities
them. In addition to including ap-
tems and number theory, computa-
for students to develop mathemati-
plied and pure mathematical prob-
tions and estimation, patterns and
cal reasoning in a variety of con-
lem types, problem representations
functions, algebra, statistics, prob-
texts. The maturation of
should be varied to provide for indi-
ability, geometry, and measure-
mathematical reasoning is a long
vidual differences, i.e., verbal, nu-
ment In reviewing what the
process. Special developmental dif-
merical, graphical, geometrical, or
Standards deem worthwhile mathe-
ferences in reasoning, especially in
symbolic, and permit several ways of
matical activities, it is important to
Grades 5-8 where students move
reaching a solution.
realize that a single assessment of
from concrete to abstract reasoners,
such activities will not provide a
must be planned for. The develop-
complete picture of a student's intel-
ment of mathematical reasoning
lectual growth. Furthermore, differ-
could be facilitated in both instruc-
Communicating
ent types of assessment are
tional and assessment settings if ap-
necessary to provide a complete pic-
propriate prompts, Why is this
Communicating about mathemati-
ture of the learner's knowledge. In
true? What if you changed this? Do
cal ideas permits students to synthe-
developing new forms of assess-
you see a pattern?, and others are
size information about the ideas.
ment one must determine the types
made available to learners.
There are a variety of modes of com-
of assessment that are best for evalu-
munication including reading, writ-
ating the various kinds of knowl-
ing, discussion, and listening or
edge. Both individuals and small
Making Connections
concrete, pictorial, graphical, or al-
groups should be assessed, but for
gebraic methods. Activities which re-
A curriculum that integrates a
different skills. Small-group learn-
quire students to communicate
broad range of mathematical topics
ing situations may be useful for
about mathematics provide them
rather than treating each topic in
measuring the ability to talk about
with opportunities to reflect on and
isolation is a connected curriculum.
and listen to ideas. Individual assess-
clarify their own thinking and to de-
Number concepts, computation, esti-
ments may be better for assessing
velop a communal understanding of
mation, functions, algebra, statistics,
the learner's ability to synthesize
mathematical ideas and notations.
probability, geometry, and measure-
knowledge.
ment become more useful to stu-
Students need opportunities to pre-
Theories of situated cognition, so-
dents when treated in an integrated
sent ideas using language to insure
cial constructivism, and the influ-
fashion. Students can be helped to
that they understand words and
ence of the group on the learning of
make connections between the top-
their definitions and meanings.
individuals can be useful in defining
ics if they are provided with con-
Teachers who structure classes to en-
authentic activities and authentic as-
texts that require their integration
courage communication provide stu-
sessment. Although research on situ-
when solving problems. It is not
dents with opportunities to validate
ated cognition is still in its infancy,
enough, however, to provide con-
their thinking about mathematics.
there is evidence that certain activi-
nections among mathematical top-
They can foster communication by
ties described by its proponents are
ics; the connection of mathematics
asking questions, posing problems,
similar_ to those described as worth-
with other topics and such disci-
or asking students to develop prob-
while by mathematical educators.
plines as science, music, and busi-
lems. Different levels of communica-
Situated cognition refers to learning
ness is also necessary (Bransford, et
tion can be obtained by interviewing
that takes place in the context in
al, 1988, Rosenheck, 1991). Teach-
individual students, by using small
which one plans to use the knowl-
ers from other disciplines can help
groups, or by classroom discussions.
edge. Problems must be realistic or
to identify the mathematical ideals
These levels permit students to ask
authentic in the sense that the appli-
that can be explored in their do-
questions, discuss ideas, offer con-
cations of knowledge are made ap-
mains. Geography, for example,
structive criticism, and summarize
parent to the learner while the
provides opportunities for the use
discoveries in writing. Cultural and
learning is taking place rather than
of scaling, proportion, ratio, similar-
gender differences should be consid-
outside of the context in which it
ity, and other mathematical ideas.
ered by those structuring activities
could be used.
Using mathematics in contexts pro-
to encourage communication.
7
Wisconsin Center for Education Research, University of Wisconsin-Madison
of dialogues that Lampert refers to
Situated Cognition
look quite different from one an-
in her work. Small groups can also.
other if they were assessed in situ-
Situated cognition has developed
foster reflection or the metacogni-
ations where limited feedback was
out of the cognitive apprenticeship
tive skills necessary to evaluate
provided in the test context. Assess-
model of instruction (Collins,
mathematical problems (Schoenfeld,
ment with feedback could measure
Brown, & Newman, 1989). The no-
the learners' potential rather than
1985).
tion of a cognitive apprenticeship
their actual performance. Learners
comes from traditional apprentice-
The theories reviewed here provide
may not need feedback the next
ships where novices learn their
great promise for building authen-
time they are tested; thus the test
trade from a master. The masters
tic activities as well as authentic as-
would have become a learning expe-
share their knowledge with novices,
sessments. There is a gap in the
rience in and of itself. This is a dy-
assisting them in developing a prod-
literature on how to operationalize
namic and adaptive form of
uct. Similarly, cognitive apprentice-
these theories. It is difficult to de-
assessment. It is dynamic since learn-
ships are designed around the
sign groups that will ensure the
ers can be retested; it is adaptive
notion that skilled learners can
sharing of cognition and optimize
since learners can learn from the
share their knowledge with less
learning for each group member. If
test. Dynamic forms of assessment
skilled learners to accomplish cogni-
more capable peers assist the less
can provide feedback to learners,
tive tasks. Cognitive apprentice-
able learners by articulating their
giving them ways to improve them-
ships, however, must model
cognitive processes, we need to
selves and opportunities to reach
cognitive processes that are often
know how to design problem solv-
their potential. Tests that serve a
difficult to externalize so that nov-
ing siutations that will allow for the
learning function may also improve
ices can observe or reflect upon the
articulation of such processes, yet
learners' motivation and sense of
skills for a particular domain. In the-
provide opportunities for the less
self-efficacy.
ory, the cognitive apprenticeship
skilled to participate in the overall
models offer suggestions for which
Social Constructivism
task.
skills to model for novices, how to
The cognitive apprenticeship model
provide scaffolding or assistance to
Authentic Assessment
is similar to the theory of social con-
less skilled learners, and when to
Authentic assessment must take
structivism (Vygotsky, 1978). Learn-
fade such assistance when learners
place in the context of the learning
ing occurs, according to the theory,
demonstrate they can construct
process. It must consider both the
when one shares cognitions with
their own meaning. Since the
learning and the situation in which
more capable peers. The NCTM
NCTM Standards call for an integra-
the learner is assessed. Authentic as-
Standards emphasize learners con-
tion of instruction and assessment,
sessment must provide information
struction, verification, and revision
the cognitive apprenticeship model
on what the learner knows or does
of mathemtical models. They also
has promise. It provides learners
not know and the developmental
stress the importance of fostering
with ways to self-reflect and correct
changes in such knowing. Repeated
problem solving, communicating,
their performances based on assess-
measures of appropriate learning in-
reasoning, and making connections
ment feedback. This theory does
dicators must be made in order to
through small group or whole class
not provide specific guidelines for
obtain a robust picture of the
discussions. Situated cognition and
when and what type of feedback to
learner's knowledge. These indica-
social constructivist theories fit the
offer or when to drop back on the
tors must include a range of cogni-
NCTM Standards well.
amount of assistance provided. If
tive and conative abilities so that
this theory were used to define
Several researchers have examined
multiple perspectives are available
authentic assessment in an opera-
the construction of mathematical
for a particular area.
tional way for mathematics knowl-
meaning using small groups (Lam-
edge, then such criteria would have
Authentic assessment will require in-
pert, 1990: Resnick, 1988; Schoen-
to be developed.
struments that provide in-depth per-
feld, 1985). Lampert discusses the
spectives on learning. Collins,
importance of finding a common
Scaffolding or adaptive feedback is
Hawkins, and Frederiksen (in press)
mathematical language for learners
important in instruction and assess-
have begun to address the best tools
to use when communicating ideas.
ment. Vygotsky (1978) proposed
for obtaining these perspectives.
The group helps facilitate reasoning
that assessment consider both an in-
They suggest that one picture does
about mathematics. Resnick de-
dividual's actual development or
not mean a thousand words when
scribes the importance of viewing
performance on a task without feed-
assessing what learners know. At
mathematics as an ill-structured dis-
back and their potential develop-
least three different assessment me-
cipline where problem repre-
ment or performance on a task with
diums, they suggest, ought to be
sentations can be discussed and
feedback during test taking. With
used to obtain an integrated picture
argued before mathematical proce-
traditional assessment where learn-
of the learners. The benefits of such
dures are employed. Resnick is par-
ers' actual development is assessed,
mediums as paper and pencil,
ticularly clear on the necessity of
it would be difficult to differentiate
video, and computers jointly pro-
having a common core of knowl-
between two learners who have the
edge in order to promote the types
same score. The two learners could
National Center for Research in Mathematical Sciences Education
8
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