ERIC ED449015: Building a Foundation for Learning Algebra in the Elementary Grades.

10 Pages·2000·0.45 MB·English

by  ERIC

DOCUMENT RESUME
SE 064 363
ED 449 015
Carpenter, Thomas P.; Levi, Linda; Farnsworth, Valerie
AUTHOR
Building a Foundation for Learning Algebra in the Elementary
TITLE
Grades.
Wisconsin Univ., Madison. National Center for Improving
INSTITUTION
Student Learning and Achievement in Mathematics and Science.
Office of Educational Research and Improvement (ED),
SPONS AGENCY
Washington, DC.
2000-00-00
PUB DATE
8p.; Published quarterly.
NOTE
R305A960007
CONTRACT
_NCISLA, University of Wisconsin, Wisconsin Center for
AVAILABLE FROM
Education Research, 1025 W. Johnson Street, Madison, WI
53706. For full text: http://www.wcer.wisc.edu/ncisla.
Non-Classroom
Collected Works - Serials (022) -- Guides
PUB TYPE
Descriptive (141)
-- Reports
_(055)
In Brief; vl n2 Fall 2000
JOURNAL CIT
MF01/PC01 Plus Postage.
EDRS PRICE
Abstract Reasoning; *Algebra; *Arithmetic; Educational
DESCRIPTORS
Change; Elementary Education; *Elementary School
Mathematics; *Mathematics Instruction; *Professional
Development; Teaching Methods
ABSTRACT
"In Brief" focuses on K-12 mathematics and science research
seeking to
and implications for policymakers, educators, and researchers
highlights the learning
improve student learning and achievement This brief
study in Madison,
gains of 240 elementary students involved in a long-term
arithmetic in ways that
Wisconsin and their remarkable ability to reason with
summarized in this
build their capacity for algebraic reasoning. The research
development,
article shows that with appropriate teacher professional
in ways that both
teachers can learn to help students learn mathematics
foundation for
enhance their understanding of arithmetic and provide a
learning algebra in
learning algebra. Teaching and policy considerations for
the elementary grades are attached.
(ASK)
Reproductions supplied by EDRS are the best that can be made
from the original document.
Building a Foundation for Learning
Algebra in the Elementary Grades
Thomas P. Carpenter, Linda Levi, Valerie Farnsworth
In Brief, Volume 1, Number 2, Fall 2000
U.S. DEPARTMENT OF EDUCATION
PERMISSION TO REPRODUCE AND
Office of Educational Research and Improvement
DISSEMINATE THIS MATERIAL HAS
EDUCATIONAL RESOURCES INFORMATION
BEEN GRANTED BY
CENTER (ERIC)
This document has been reproduced as
received from the person or organization
S. Anderson
originating it.
O Minor changes have been made to improve
TO THE EDUCATIONAL RESOURCES
reproduction quality
INFORMATION CENTER (ERIC)
Points of view or opinions stated in this
document do not necessarily represent official
OERI position or policy.
1
AVAILABLE
BEST COPY
MATHEMATICS & SCIENCE
NATIONAL CENTER FOR IMPROVING STUDENT LEARNING & ACHIEVEMENT
IN
K-12 Mathematics & Science
,nBtriet
RESEARCH & IMPLICATIONS
FOR POLICYMAKERS, EDUCATORS & RESEARCHERS
SEEKING TO IMPROVE STUDENT LEARNING & ACHIEVEMENT
Va.
k
A
-7
-7
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ABOVE: Elementary students-wojik-onsolving mathematics probleins and developing generalizations.
BUILDING l'As
A
research,1 Carpenter and Levi support
%):116 study led by researchers Thomas
teachers' professional development by sys-
ter and Linda. Levi citthe 'National .Center
tematically helping them to,,focus on students'
g and
for.'Impiiiving: Student
mathematical thinking; in particular students'
inMathernatics,,-arid
FOR LEARNING
,
7r
abilities to .articulate, represent, and justif)F1
Science ( NCISLA), found that
.
generalizationS about the underlying structure
innovatiire, teacher
;
and properties of arithmetic:
sionallievelopinent and refb-
, rnsed-7;mathernatitVinstinc-
,.,,--
Building on student thinking.,The researchers
tion paved the way for even
and teachers have:found that students have a
7.: "first- and cfsuec-aridlgiaders to
great deal of implicit knowledge about-basic
begin to reason algebraically.
tementctry
properties of arithmetic. For example, even
-
--Research results': show :that
1N- :THE
first-grade students: will' choose to count on
young students can learn to
.
from the larger number to find a sum like 3+
make and justify:generalilations
9:
These students implicitly recognize they can
ahout the underlying structure and
interchange the order from 3'4: 9 .to 9 + 3 to
properties_of.arithmetic
tV
easier: _When these basic
make the
tions that form-the basis for much Ofilge-
,
braterit-l with goals
in the
,p_r_op_erties are not made .explicit, however,
in
many students are unsurewhether the proper -
-Nationalcouncil of Teachers of Mathematics
rr eachers.,and-;researcheg have long recog
_..ties apply-in-new or unfamiliar problem con-
nized that the-transition' from learning
.'(NCTM) Principles,And-Stindards (2000),. this
texts. For example, .students may not under-
arithmetic hi' learning: algebra is one of-the'' NCISLA study.shOws_that
stand that they can similarly change the order
major hurdles studerifs*einlearning math-
begin bniging.a_fon_ndation.inalgebra much-
of the numbers if they are adding very large
.earligi than typical curricula allow.
.eniaticsAtesearchers,3are finding, however,
numbers, or fractionS, or expressions involv-
ir4c well/
thatelementary stUdehts 'can learn to think
-
":.=.
r
.
.
ing variables in, algebra. Other students may
,both enhance
Focus
abchitaffifunetic iin
,RESEARCH
overgeneralize:.They may for example, inter-
their early learningof ar*ietic_ancl provide
Reieciith--Project
''.-The Tiifiy-.-Algebra
change the order of numbers when subtract-
forleargiiiialgebra. This in Brief
a foundation
.
ing or dividing.
ird year, the early alge-
highlights learning gains of 240 elementary
iirrefitly. in its.
students' involved in a long-term- study in
is led by NCISLA researchers
%;,,,:ibia:piciject
Algebra builds on the same fundamental prop-
Carpenter and Levi and 12 Madison
Metro-
Madison,' Wisconsifif,1 and their remarkable
.
erties that form the-basis for arithmetic. The
Politan School District teachers. Building, on
ability to reason-about arithmetic in ways that
abstract nature of algebra makes it even more
15 years 'of Cognitively Guided Instruction
build their capacity for algebraic reasoning.
1 The research in Madison is peridding a foundation for teacher professional development in schools in Phoenix, Los Angeles, and San Diego.
own content
2 The Cognitively Guided Instruction Professional Development Program engages teachers in learning about the development of children's mathematical thinking in particular content areas, building their
1999.
knowledge, and refining instructional practice. For more information about CGI, see in Brief reference: Children's Mathematics: Cognitively Guided Instruction (a book and two multimedia CDs),
.
1
eralizations from students in the class. The
Yeah. 49.78 49 + 49 = 78.
important that students understand precisely
MIKE:
class then discussed these generalizations and
when and why properties of arithmetic can be
Do you all think that is true?
TEACHER: WOW.
whether they were always true for all num-
[All but one child answered yes.]
applied. The goal of the early algebra project is
bers, which led to an extended classroom
to help students make explicit and understand
I do, because you took the 49 away,
JENNY:
analysis of what is required to justify a gener-
the underlying structure and properties of
and it's just like getting it back.
alization. This form of instruction built on
as they are learning arithmetic
arithmetic
[Emphasis added to lift out teacher's question strategy.!
student thinking and supported their under-
so that they will have a solid base to build on as
standing of basic properties of arithmetic
they go on to learn algebra with understanding.
As the discussion above continued, the group
required for algebra. The following classroom
excerpts illustrate how these goals were
collectively came up with the generalization:
Supporting professional development. A criti-
"Zero added to another number equals that
accomplished.
cal component of the early algebra project is
other number:' They also came up with the
teacher professional development. At summer
First- and Second-Graders'
generalizations: "Zero subtracted from another
workshops and meetings held throughout the
number equals that number," and "Any num-
Capacity for Algebraic Reasoning
year, the teachers and researchers analyzed the
ber minus the same number equals zero." The
structure and basic properties of arithmetic
students not only applied these generaliza-
tudents from a first- and second-grade
and considered learning contexts that could
S
tions to solve problems involving zeros, they
class were given a set of problems to guide
encourage students to explicitly articulate gen-
also came up with number sentences that
them into expressing a generalization about
eralizations about these properties. The group
embodied more complex ideas (e.g., 78 49+ 49
what happens when zero is added to a number.
considered how students might think about
The children were not only able to articulate
= 78).3
specific problems and ways students might jus-
the generalization; they were able to take the
tify (or prove) their proposed generalizations
The study shows that even first- and second-
discussion to a higher level mathematically.
bet-
were true. These sessions helped teachers
grade children are able to argue about mathe-
ter understand their students' thinking and
matical concepts and operations in ways not
The teacher asked the students if "78
49 = 78"
build their own mathematics knowledge. In
generally expected of students at this age.
was true or false. The students immediately
addition, the meetings supported the evolu-
These students were able to express generaliza-
responded:
tion of a committed and growing professional
tions and reason about them.
community.
False! No, no false! No way!
CHILDREN:
Why is that false?
Third- Through Fitth-Graders'
TEACHER:
Enhancing classroom instruction. True-false
Because it is the same number as
Introduction to Mathematical Proof
and open number sentences were the primary
JENNY:
in the beginning, and you already
means of eliciting generalizations from students.
took away some, so it would have
hird- through fifth-grade students partici-
(See "Number Sentences Used to Generate Gen-
to be lower than the number you
pating in the early algebra project not
eralizations".) Teachers used true-false sentences
started with.
only identified more complex generalizations,
to initiate discussions that focused on how stu-
Unless it was 78 0 = 78. That
they also were challenged to justify their gen-
MIKE:
dents knew a sentence was true or false. This
would be right.
eralizations using arguments that helped them
method generally was sufficient to elicit gen-
to gain an appreciation for mathematical
Is that true? Why is that true? We
TEACHER:
NUMBER SENTENCES USED
took something away.
proof. They learned that justification went
Generalizations
to Generate
beyond proposing an endless supply of exam-
that something is, there is,
But
STEVE:
that
like, nothing. Zero is nothing.
ples for which the generalization applied.
Below are examples of number sentences
teachers used to help students articulate gener-
Is that always going to work?
TEACHER:
alizations about zero and multiplication.
For example, Mary Bostrom's third- and fourth-
If you want to start with a number
LYNN:
Examples: 78 + 0= 78; 23 + 7 = 23'
grade students were asked to justify the gener-
and end with a number, and you
"When you add zero to a number,
alization: "When you multiply two numbers,
do a number sentence, you should
with."
you get the number you started
you can change the order of the numbers" (ax t
always put a zero. Since you wrote
=74
Examples: 96 96 =0; 74
= tx a). The students initially calculated a lot of
78 49 = 78, you have to change a
49 to a zero to equal 78, because if
examples, such as 8 x 5 = 5 x 8, but Ms. Bostrom
"When you subtract a number from itself,
you get zero.'
you want the same answer as the
pressed them to show that the generalization
first number and the last number,
was true for all numbers, not just some.
Examples: 96 x 0 =0; 43 x 0=43'
you have to make a zero in
"When you multiply a number times zero,
between.
To prove the generalization was always true,
you get zero.'
do you think that will always
one pair of students went back to the basic def-
TEACHER: So
Examples: 65 x 54 = 54 x 65; 94 x 71 = 71 x
work with zero?
inition of multiplication using linking cubes
"When multiplying two numbers, you can
Oh, no. Unless you 78 minus,
to illustrate a specific example (8 x 5 = 5 x 8, see
MIKE:
change the order of the numbers."
umm, 49, plus something.
Figure 1). After some discussion, the students
denotes a false number sentence
provided a concrete justification, demonstrat-
Plus 49.
ELLEN:
After students were introduced to variables and open number sentets, they were able to express generalizations in open number sentences that
3 Students at first used everyday language to express generalizations.
were always true, such as b-b=0,andb-a.a=b.
BEST COPY AVAILABLE
algebra can be a part of elementary instruction
calculation or that they should just add all the
students'
FIGURE 1. Proving
8 x5 = 5 x8: Two
that builds on students' implicit mathematical
numbers together. Furthermore, teachers par-
proof using the rotation of cube arrays.
knowledge and increases their ability to
ticipating in the research found that explain-
understand, reason, and engage in challenging
8x5
ing the equal sign was not sufficient to ensure
problem solving.
that students understood its meaning. Build-
ing off students' different conceptions of what
5x8
The instructional strategies outlined here have
the equal sign meant, teachers engaged stu-
significant implications for teacher learning
dents in mathematical discussions that helped
and professional development. Through their
them confront their misunderstandings and
ongoing work with fellow teachers and
achieve an accurate understanding of the mean-
researchers, the teachers gained essential
ing of the equal sign.
insight into their students' mathematical rea-
(b)
(a)
soning; they also forged a community through
NUMBER SENTENCES THAT
Equality
Ccncepticn
which they gained mathematical knowledge
Challenge Students'
.
.
and crafted problems that benefited their stu-
ing that they could rotate one group of cubes
7=3+4
a)
dents' learning. Although it requires a signifi-
(Fig. la) 90 degrees and place the rotated array
8=8
schools
cant commitment and support from
b)
"Look, they are
over the other group (Fig. lb).
and policymakers (see insert on Policy Consid-
5+8=8+5
the same," exclaimed one student. The other
c)
erations), participants in both CGI and the
student added, "That works, but you don't
8=5+13*
d)
early algebra project have seen this form of
[Fig. lb). You
even have to have the other one
denotes a false number sentence
exciting
professional development yield
and see the
can just turn this one [Fig. la]
results in students' learning of mathematics.
other groups."
Interested educators should also refer to
Across several class periods, the teachers con-
the Teaching Considerations insert, which
tinued to provide examples of true and false
Although the students used a specific example
specifies some resources and strategies
number sentences that challenged students'
they
to justify the generalization, the way
teachers might adopt with their students and
conceptions of the meaning of the equal sign
explained the example showed that they
colleagues in their educational community.
and reinforced their learning. These discus-
understood that they could do the same thing
sions resulted in significant changes in stu-
with any numbers. These students were begin-
For More Intormation
dent understanding and problem-solving
ning to learn what was required to justify that
improvements (see Figure 2).
all numbers.
a generalization was true for
A research report about the early algebra proj-
listed
ect and other relevant publications are
Equality
of
IMPLICATIONS:
Students' Understanding
in the reference section of this in Brief and are
Elementary
Retiorm
available at the NCISLA website at http://
Mathematics inatruction and
equality and the
An understanding of
Teacher Proteadional Development
www.wcer.wisc. edu/ncisla/.
appropriate use of the equal sign is criti-
cal for expressing generalizations and for
Researchers Tom Carpenter and Linda Levi
T
his study showed that young students can
developing algebraic reasoning. For example,
National Center
can be reached through the
learn arithmetic in ways that provide a
the concept of equality (indicating a relation-
for Improving Student Learning and Achieve-
foundation for learning algebra. Recognizing
ship between different parts of an equation
ment in Mathematics and Science at the Uni-
young students' ability to reason algebraically
and meaning "the same as") is embedded in a
versity of Wisconsin-Madison, 1025 W. John-
does not suggest that elementary students
8. Unfortu-
5= 5
number sentence like 8
x
x
263-3605;
son St., Madison, WI 53706; (608)
should learn high school algebra. Rather, this
nately, many students typically hold miscon-
E-mail: [email protected]
study showed that a broader conception of
ceptions about the meaning of the equal sign
(see Kieran, 1981; Matz, 1982) and tend to think
FIGURE 2. Students' Increase in Understanding of the Meaning of the Equal Sign
it means merely to carry out an operation.
In order to assess students' understanding of
; ooZ
got
the meaning of the equal sign, the researchers
Percent of students
correctly answering
and teachers gave the students a seemingly
8oR.
8+4. +5
228
-66Z
70Z-
simple number sentence:
before and after teachers
6oZ
adjusted instruction.
8+4=0 +5
soZ.
40X
Consistent with previous research (Kieran,
30%
Before Adjusting
1981; Saenz-Ludlow & Walgamuth, 1998),
20Z
Instruction
......
most students responded that either 12 or 17,
9%
loZ
sz
ox
After Adjusting
rather than 7, should fill in the box. They
oX
Instruction
Grade 6
Grades 3 & 4
Grades ; &
2
thought either that the number immediately
(N78)
(N.19)
(N -/13)
after the equal sign had to be the answer to the
5
BEST COPY AVAILABLE
References
ABOUT in Brief
Carpenter, T. P., & Levi, L. (2000). Developing con-
and
This publication and the research reported herein are supported under the Educational Research
ceptions of algebraic reasoning in the primary
of
Development Centers Program, PR/Award Number R305A960007, as administered by the Office
grades (Res. Rep. 00-2). Madison, WI: National
Center for Improving Student Learning and
Educational Research and Improvement, U.S. Department of Education. The opinions expressed
the
Achievement in Mathematics and Science. [Avail-
herein are those of the featured researchers and authors and do not necessarily reflect the views of
able at www.wcer.wisc.edu/ncisla]
supporting agencies.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L.,
Empson, S. B. (1999). Children's mathematics:
Permission to copy is not necessary. This publi-
111ARIgki-4.4
fp:al
Cognitively Guided Instruction. Portsmouth, NH:
cation is free on request.
K-I2 Mathematics & Science
Heinemann.
inBriely
RESEARCH & IMPLICATIONS
in Brief Volume 1, Number 2 is a publication of
RESEARCHERS
(LOS
FOR POLICTORAKERS. EDUCATORS
Carpenter, T. P., & Franke, M. L. (1998, fall). Teachers
A0.1.11110,
SEA
SURMA 10 01111010 STUOINT
as learners. Principled Practice, 2 (2). [Available at
the National Center for Improving Student
www.wcer.wisc.edu/ncisla]
Learning and Achievement in Mathematics and
Falkner, K. P., Levi, L., & Carpenter, T. P. (1999).
Science, at the Wisconsin Center for Education
Children's understanding of equality: A founda-
Research, University of Wisconsin-Madison.
tion for algebra. Teaching Children Mathematics, 6
10. *
(4), 232-6. [Available at www.wcer.wisc.edu/
...
BUILD MS
A
1
Foundation
1
Kieran, Carolyn. (1981). Concepts associated with the
sImmatolool
.
&Hamm l Whom/.
FON
IMG
air
NCISLA Director: Thomas P. Carpenter
equality symbol. Educational Studies in Mathe-
kims 01CUL/111 Oa.
exA re,
104.0*
matics 12,317-26.
.............
NCISLA Associate Director: Richard Lehrer
.1
ad um.= Ilo.1. De ...A..
ai
. a =Oa
Matz, Marilyn. (1982). Towards a process model for
Ws b............:
Elementary
osigs O.. ua
h amp..
REIgal Relmea
IN TNN
rnim
school algebra errors. In D. Sleeman, & J. S. Brown
Amanse,...1
cm Imo le
NCISLA Communication Director:
win aelltiksowIlmo
Grade
a
De mama web. Room. a..
(Eds.), Intelligent tutoring systems (pp. 25-50). New
Mil.. NW
11.-
e Mom as woe
Susan Smetzer Anderson
.1,1* bat
b.. al. wel
York: Arademic Press.
Mama or
no
Nob .1.ser
mu,
41.1
sara.
ena=or.
aaa.
.1 1.1/1..0116
Ns... mem
1 awe
um. gius
INCTIO
ca
Writers: Thomas P. Carpenter, Linda Levi,
National Council of Teachers of Mathematics. (2000).
Orimla
tb
arbes
Saly Maw shalom blessma
armpit dana.*
Principles and standards for school mathematics.
maaSL Mambas WO ray rata.
ORA imOre b WI.
Valerie Farnsworth
a tram. mo saws
4.1. m.o. um WAN
Oil.*
Vegada
Yob ahet
Am. andrams in won
.1, 99 KOlf
all....easia ad*,
Reston, VA: Author.
=la ."."*"...
an.. is Ion.
irskr. be.
mid* a . 1
so *moan
Designer: Kristen Roderick
Saenz-Ludlow, A., & Walgamuth, C. (1998). Third
14/1 Km.; AY !I. /. a,/ I! Man 1.28. otal.*111%
Gam.
graders' interpretations of equality and the equal
Av.. Rm... Icamaa 716.
'ALM Arbal [VIM *en
Cos.. Caled Wawa. lama alt or.
.1HDRame, ix *um Ammo
if wan
symbol. Educational Studies in Mathematics,
35q(2),
153-87.
ED
Nonprofit Organization
NATIONAL CENTER FOR IMPROVING STUDENT
U.S. POSTAGE
LEARNING & ACHIEVEMENT IN MATHEMATICS & SCIENCE
PAID
Wisconsin Center for Education Research
Madison,Wisconsin
School of Education, University of Wisconsin-Madison
Permit No. 658
1025 West Johnson Street, Madison, WI 53706
H1OU1 BECKRUM
DATABASE COORD
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.c7.25
77Z.'1*
STUDENT LEARNING & ACHIEVEMENT IN MATHEMA
NATIONAL CENTER FOR IMPROVING
K-12 Mathematics & Science
inBriet
POLICY CONSIDERATIONS
Elementary Students Can Reason Algebraically
dents' ways of solving group-selected mathe-
he research summarized in this in Brief.
The
matics problems and number sentences.
1 shows that with appropriate teacher pro-
learn-
teachers themselves were learners"
learn to
fessional development, teachers can
_learning
ing about students' thinking and
help students learn mathematics in ways that
'eiders can
mathematics. School and district
both .eilhance :their:understanding of arith-
foster the development of these kinds of com-
metic and provide a foundation for learning
teachers
munities by making time available for
algebra. Policymakers and administrators can
for teach-
to meet, by arranging opportunities
kind ._.of
support the development of thiS
planning time_as they
ers to have common
instruction in_ their schools by
and by
work to develop such a community,
being active participants in the communities.
-e Supporting sustained, long -term programs
Of professional .deirelopment that focUs on
thedevelopment of t-uderits' mathematical
Focusing on Students'
-
ABOVE:--- Elementary students reason together
Shared Vision
thinking; as well as teachers' understanding
Mathematical Thinking
about mathematics problems.
.'.'ofihe related mathematical ideas:
chool and district leaders can help teachers
.profes-
e Supporting the development.of
S
focus on developing students' mathemati-
ing and how it is reflected in their classroom
sional communities that help teachers
in
cal thinking by making that focus a priority
-interactioni-iiith-sludents, as well as time to
;make their Classiocimi_placet 'for both
their
the school. By talking to students about
reflect on their teaching and their students'
dents and teachers to engage in inquiry and
mathematics and by talking to teachers about
understandings andmisconceptions. Teachers
learn with understanding.
princi-
their students' mathematical thinking,
also need access to resources; including people
ez.gncouragnig.teachers to conceive of teach-
which
pals can help create an atmosphere in
and materials, that can help them develop
ing in terms of understanding and develop-
One
such conversations become the norm.
their-understanding.
ing students' Mathematical thinking.
principal in the project asked teachers to regu-
work to
larly bring examples of their students'
School. and district-leaders can also support
students
her and discuss with her what the
professional development by participating in
Suatained,Long-
Supporting.
understood and were learning. Teachers'
it themselves. Leaders need to understand the
Term Protesaional Development
analysis of their students' thinking was
ideas that the teachers are learning so that they
included as a significant part of the teachers'
can understand whatds going on in the teach-
here are no simple solutions foithe prob-
candidates'
annual 'review. Questions about
ers! classrooms and-provide the necessary sup-
a- lems facing our schOok Simply introduc-
understanding of and interest in understand-
port and assistance when called on. Further-
iiig-Tiiiiiitirticulunrorproviding teachers a
might
ing students' mathematical. thinking
more, the participation of all interested parties
one-week workshOp. i..9iotIgoing to resultin
of hiring
also be an important component
communicates..to. the -teachers that their pro-
sustainable improvement_ in teaching or sig-
interviews.
fessional development: is important and . val-
nificant gains-in student achievement. Fonda-
--
ued by the school_ and district.--
change requiresthat teachers develop a
- mental
deep understanding ofthe matheinatics they
For More Intormation
Supporting Prote&sional Communities
teach and the ways their students think about
and learn that mathematics. Developing that
Policymakers who would like more informa-
iacheis- participating in the NCISLA
T
understanding takes time and requires sus-
sustainable
tion about student learning and
early algebra project entered into the
tained, long-term prafesSional development.
teacher professional development can contact
project at varying levels of knowledge and
School and district leaders need to find ways to
Student
the National Center for Improving
matEematics 'confidence; they also came into
provide resources to support those kinds of long-
Mathematics
Learning and Achievement in
it from different types of schools and class-
term professional development opportunities.
and Science (NCISLA) at [phone] or
_
student
rooms (in terms of grade level and
.
www.wcer.
refer to the NCISLA web site:
diversity). Together with the researchers, these
One of the most critical resources is time.
wisc.edu/ncisla. Researchers Tom Carpenter
teachers committed to forming a professional
Teachers need time to participate in the pro-
the
and Linda Levi can also be reached at
development community. They met every
fessional development, time to meet with
NCISLA.
stu-
month and focused their discussions
other teachers to discuss what they are learn-
.
I
.
k
LEARNING SI ACHIEVEMENT
IN
al/NATIONAL CENTER FoR IMPROVING STUDENT
K-i2 Mathematics & Science
inBriet
TEACHING CONSIDERATIONS
Algebraically
Elementary Students Can Reason
,-------
;
early-algebra-projea committed-to
.
NCISLA
NUMBER SENTENCES
Brief
rr he research, summarized in this in
forming a professional development corn-
to Elicit Generalizations
1 showed that elementary students can
miuulk.Thei met every month and focused
false?
learn to think:about arithmetic in ways that
Is this number sentence true or
mathematical '-
their discussions on students'
both enhance their learning of arithmetiC and
48x0=48*
how
97 97 = 0
thinking, ways to elicit that thinking,
56 +0 =0"
provide a foundation for learning algebra. If
37+58=58+37
to figurnnupshatstutilents
ideas in your class-
you want to applk..these
What number can you put in the box
IR discussing math-
ways to engage students
to make this a true sentence?
following:
room, consider the
ematicif ideas. The teachers themselves
35 x0=8x35
0+74=74
were.- learners -
Build a Foundation
thizikingand-4eamini about mathematiis-
denotes a false number sentence
or Learning Algebra
window into
ei Ask questions that provide a
For More bitormatiOn
bei from itself, you get zero." When-they do
children's-understanding .of important
&p.
state a generalizationlilsOlics ask, for exam-
mathematical ideas. For example, students:.
publications
Elementary teacher resources and
-pie, Is that true for alfnumbesr.,.:!
sentence
responses to the number
+
www.wcer.viise.eduincislal:Sier
are available at
their.iinder-
e Have-students justify generalizationithey:or
0 + 8" tells -a. great deal about
1
4e-vilot)-
standing of the meaning of the equal-sign,
Teacher learpial and. pzbfesSi anal
theiL'peerse.projiise (see-page 3 for an- exam -.
1998 newsletter
Prcibe itudenl's -feisito for: their answers.
ple). Justification of-generalizations re*
ment:- "-The---NCISLA
Principled. Practice: Teaikeis -as-Lek-ruffs,- at
Ask studentsohy they answered as they did.
(e.g.,
.more than providing.a lot Of examples
www.wcer.wisc.eddicisla.
By expecting-children to justify
x 5= 5
8
x 8).
discuss
e Provide students opportunity to
Ckik.
Children's understanding of equality:
stalls, in
you can help them gain
their
claims,
and resolve different conceptions of mathe-
Foundation
pre-senting mathematical arguments and
dren's Ilndemndirs ofAquality: 7,A,
matical ideas. For example, the different
Linda Levi iad
Karen-P:
proofs. Use the:questions "Will that be true
for Algebra,
conceptionS.Of 'the equal sign that emerge
Children
ThomaS..P. Carpenter,: in-:Teaching
for alLnumbersMantF"How do you know
from students' solntions to the open num-
DeC..1999.- (Also-at
Mathematics, .vol. 6,-
that is true for all numbers?"--repeatedly
ber sentence '9 + 6 = 0 + 8!. can provide the
www.wcermisc.ecluinclila/teichers)
they
encourage students to recognize that
basis for a productive discussion._
dithstiiii's
Cognitively Guided-CrisI:ruction:
in mathematics.
need to justify their
claim's
that help
e Provide students with equations
Instruction
Mathematics - Cognidiely Guided
Car:
them understand that the equal sign repre-
(with t*Itiultimedia CDs),.% Thomas. P..
Form a TeaChing.TCommunity Focused
Feiineina, . Megan Loef
snits a relation between numbers, not a sig-
penter,. Elizabeth
on Students' Mathematical Thinking
Franke, Linda Levi, -and'sSusan B. Empson.
nal to carry out' the preieding calculation.
:1_
Publishers;1999.
e Make classrooms a place where both you
Heinemann
Examples include ff0= 8 +9"'"8 +6 =6 +0,"
and your students are learning. Engaging
87.Varythe format of number sen-
"9 + 6= 0 +
in regular inquiry about students' mathe-
tences: Include sentences in which the answer
matical thinking can be one of the most
does not come right after the equal sign.
Mt
powerful forms of professional develop-
false num-
e Provide students with true and
ment. Such -inquiry' does not,. Jaowever,
ber sentences that challenge their miscon-
thrive inisolation. Seek out other teachers
ceptions about the equal sign (e.g.,
8 = 5 + 3,
who share your-interest in talking about stu-
9=9, 7-4=7111).
dents' thinking and share with one another
s Provide students problems that encourage
the interesting observations you are making
them to make generalizations about basic
about students' mathematical thinking in
number properties (see "Number Sentences
your
classes.
they pro-
to Elicit Generalizations".) When
Form a community with teachers and
e
vide an answer to one of the problems, ask
other resource people. You will find the
them how they know their answer is correct.
the
Together . with
support invaluable.
elementary student explains his rea-
ABOVE: An
That often will result in their stating a gener-
soning to his teacher and fellow students.
researchers, the teachers participating in the
alization such as "When you subtract a num-
8
I
I I
002
NCISLA
MON 14:05 FAX [phone]
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S(o'S
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Title: Building a FOundation for Learning Algebra
Valerie Farnsworth
Author(s): Thomas P. Carpenter, Linda Levi,
Student Learning and Achievement (NCISLA) in
Corporate Source: The National Center for Improving
Mathematics and Science
Publication Date: January 2001
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