ERIC ED480462: Algebraic Skills and Strategies for Elementary Teachers and Students.

13 Pages·2003·0.4 MB·English

by  ERIC

DOCUMENT RESUME
ED 480 462
SE 068 318
Algebraic Skills and Strategies for Elementary Teachers and
TITLE
Students.
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.
PUB DATE
2003-00-00
NOTE
12p.
R305A960007
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AVAILABLE FROM
National Center for Improving Student Learning and
Achievement in Mathematics and Science, Wisconsin Center for
Education Research, University of Wisconsin-Madison, School
of Education, 1025 W. Johnson Street, Madison, WI 53706. Tel:
608-263-7582; e-mail: [email protected]; Web site:
http://www.wcer.wisc.edu/ ncisla.
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Guides
(055)
In Brief; v3 n1 Sum 2003
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DESCRIPTORS
*Algebra; Arithmetic; Cognitive Processes; Elementary
- Education; *Elementary School Mathematics; Elementary School
Students; Elementary School Teachers; Mathematics
Instruction; Professional Development; Teacher Improvement;
Thinking Skills
ABSTRACT
Research shows that introducing basic forms of algebraic
reasoning in elementary school enhances children's learning of arithmetic. In
this guide, a research and a professional development project are described
that have taken a practical approach to introducing algebraic reasoning to
elementary students in Massachusetts. (Author/SOE)
Reproductions supplied by EDRS are the best that can be made
from the ori inal document.
[Algebraic Skills and Strategies for Elementary
Teachers and Students]
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
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Points of view or opinions stated in this
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official OERI position or policy.
2
BEST COPY AVALABLE
,
K-12 Mathematics & Science
inBriet
RESEARCH & IMPLICATIONS
FOR POLICYMAKERS, EDUCATORS & RESEARCHERS
SEEKING TO IMPROVE STUDENT LEARNING & ACHIEVEMENT
/0"1111111.
_Asa ____ffr4tg
Algebraic
into
teachers'
reasoning
algebraic
If 5 people in a group shake hands with each
current lessons as early as the first grade.
other once, how many handshakes will there
Instead of treating algebra as an add-on
be? What if there are 6 people in the group?
Skills & Strategio tor
7 people? 8 people? 20 people? Write a
to the curriculum, the researchers rec-
ommend "algebrafying" the curricu-
number sentence that shows the total number of
Elementary
lum, an approach that takes into
handshakes. How did you get your solution?
account teachers' challenging work
Show your solution on paper. (See box on page
environment as well as traditional
4, Focus onthe "Handshake Froblem':)2
Teachens
elementary teacher preparation. A
"Jan",3 the case study teacher, used this prob-
case study of students' learning,
lem as a basis for a lesson for her third-grade
&Studenta
including data on students' perform-
students. After dividing her students into
ance on the Massachusetts Compre-
groups, Jan asked them to act out the
hensive Assessment System (MCAS), pro-
problem and determine the total number
I
n the United States, students typically are
vides preliminary evidence that this strategy
of handshakes in each group. She directed
not introduced
algebra until
the
to
learning and
improves young students'
students to consider helpful ways to keep
8th-lOth grades, but research shows that
achievement in mathematics.
number of
their
track
data (the
of
introducing
forms algebraic
of
basic
handshakes) and to model their process
reasoning in elementary school enhances
"Algebratying" Elementary
arithmetically. Students formed number
learning arithmetic.
The
children's
of
Mathematio Inotruction
National Center for Improving Student
sentences such as 7+ 6 + 5+ 4 + 3+ 2+ 1, found
ways to group numbers (e. g., grouping the
Learning and Achievement in Mathematics
number of handshakes occurring when
and Science (NCISLA) has found that
he goal of this Massachusetts project has
T
there were 5 people in a group and then
young children can learn to reason alge-
been to treat elementary school mathe-
counting on to determine the number for a
matics, especially arithmetic, in a more
braically) Described here is a research and
way. Jim Kaput and Maria
group of 8 people), and identified patterns
professional development project that has
algebraic
that ultimately led them to use multiplica-
taken a practical approach to introducing
Blanton, NCISLA researchers who lead this
tion to solve the problem. By writing num-
algebraic reasoning to elementary students
project, offer the following as an example
ber sentences and then working with the
of a traditional arithmetic problem that can
in Massachusetts.
form of the number sentences (rather than
be transformed into one that develops
simply computing a number for each case),
students' algebraic reasoning and provides
As part of a Massachusetts district-wide school
the students came to work with numbers
improvement plan, a NCISLA research team
students with skills to solve new and more
and operations in an algebraic way.4
has been leading a professional development
challenging problems:
program focused on strategies to integrate
See also in Brief (2000) article on the NCISLA early algebra research project headed by NCISLA director and researcher Tom Carpenter.
3
2 For an earlier version of the handshake problem that appears frequently in professional literature, see Yarenta, Adams, & Cagle (2000).
3 "Jan" is a pseudonym.
Modeling, Generalizing,
4 A video of this classroom episode is available through Annenberg/CPB and also on the forthcoming NCISLA CD-ROM, Powerful Practices in Mathematics a- Science:
& Justifying, to be distributed by the North Central Eisenhower Mathematics and Science Consortium at the North Central Regional Educational Laboratory. (See box on page
4.)
0 I
I
B riqf : ALGEBRA IC St ,ate,6 e4 for EI e,m,e!ny,ary Grades
,At'AB L
-TI:`
._,
induded planned activities, most of which (63
examining Teacher
instances) were activities Jan developed from
How many of the smallest squares will be
Practice & Student Learning
her own instructional resources.
in Figure 5 if this pattern continues?
ra
z aput and Blanton's team conducted a
Student learning.
At the conclusion of the
year-long case study of Jan's class to
study, the researchers administered a set of
awl
document student learning of algebraic
14 test items from the fourth-grade MCAS to
reasoning. The researchers also documented
the 14 third-grade students present in Jan's
the effects of this professional development
class the day of the testand compared the
program on Jan's instructional practice as
results to a control group of third-grade
well as on student achievement on selected
students in the same school.6 The results offer
FIGURE 1. Sample MCAS problem
items from the fourth-grade MCAS (in com-
some preliminary evidence to support the
identified as "deeply algebraic':
parison to a control group and the district).
value of the algebrafication and professional
development strategy implemented in Jan's
Professional Development
Teacher practice integrating algebraic
classroom (Blanton & Kaput, in press-a).
Blanton
reasoning.
observed Jan's
for Instructional Change
90-minute third-grade math class about 3
Jan's experimental group performed better
days a week during the course of an academic
than the control group on 11 of the 14
rip he professional development project
year (38 visits total). Following aspects of
of which were
test items
selected
(4
provides teachers and administrators a
the design experiment approach,5 Blanton
significant at alpha = 0.05). Jan's students
practical approach to changing elementary
worked closely with Jan,
occasionally
scored higher than the control group on 6
mathematics instruction in ways that build
co-teaching the course. Through collecting
out of the 7 items that the researchers
students' algebraic thinking. Now in their
Jan's reflections and examples of student
identified as being deeply algebraic in
seventh year of the project, Kaput and Blanton
work and visiting Jan's classroom, the
nature. These problems, such as the one in
have worked with administrators in an
researchers documented the ways in which
Figure 1, required students to find patterns,
academically underachieving Massachusetts
Jan integrated algebraic reasoning into her
understand
whole-number
properties,
school district8 to implement a research and
(Blanton & Kaput,
instruction
2002).
and identify unknown quantities in a
professional development program as part of
number sentence.
the improvement plan required by the Mass-
The researchers were interested in the
achusetts Department of Education. The
diversity and frequency of Jan's integration of
In addition, the results from Jan's third-
program was touted as "exemplary" by writers
algebraic reasoning. They identified a total of
grade classroom also were compared to the
of the No Child Left Behind Act Summary of
206 instances of algebraic reasoning covering
performance of the district's fouth-grade
12 different types of algebraic practice.
the 2001 Reauthorization Conference Report
students on the MCAS: A higher percentage
Of these episodes, 132 (65 percent) were
of the U.S. Senate Health, Education, Labor,
of students in Jan's class scored at the
characterized as instances in which Jan
and Pensions Committee. The program
"advanced" and "proficient"
levels
(see
spontaneously crafted instruction that
addresses both teachers' and administrators'
Blanton & Kaput, in press-a). These results
required students to reason algebraically.
needs through a leadership academy and
are noteworthy given that many of Jan's
Blanton and Kaput considered this
professional development seminars.
students were from homes where English
significant, indicating that Jan's knowledge
was either not the primary language or not
The leadership academy. Kaput and Blanton
of algebraic reasoning enabled her to see
spoken at all. The socioeconomic status
consider administrative support, especially at
ways that algebra could be integrated into arith-
of students in Jan's third-grade class was also
the school level, crucial to the success of teach-
metic lessons. The remaining 74 episodes
lower than average for the district.7
ers' professional development. In order to
build the capacities of principals to support
Enhancing Student Learnins Threugh Alsebraie M6kA
instructional change, the researchers and
district superintendent held seminars for all
K-5 principals and curriculum coordinators
Algebraic tasks like the handshake problem can help students learn to
once a month for one semester. Given the
e Represent data.
principals' personnel responsibilities, the
e Construct a number sentence that models a phenomenon.
leadership academy was designed to help
e Examine how variations in a phenomenon affect the number sentence.
administrators understand the algebrafication
strategy and how that approach could inform
e Use a number sentence to reason algebraically about a problem.
hiring decisions, teacher evaluation, and day-
e Understand the properties of whole numbers and the number system.
to-day supervision. The sessions addressed
e Understand the relationships between operations in order to facilitate
practical concerns such as the implementation
computation (e.g., recognize repeated addition as multiplication).
of new state curriculum frameworks and
ways to allocate more time for teachers to
5 For more on design experiments, see Cobb, Confrey, diSessa, Lehrer, 0- Schauble (2003).
6 The teacher in the conta.ol classroom did not participate in the professional development program.
7 About 75% of the students in Jan's class were on free lunch, and 15% on reduced lunch; 65% were from families for whom English
was a second language; 25% had no parent living at home.
s The district includes 30 relatively small elementary schools.
0
a
f
4.
a
;it
RESOURCES
for
TEACHERS
POLICYMAKERS
Developing Children'a
Algebraic Recooning
thinking
issue
focus
Algebraic
[Special issue]. (1997). Teaching
Children Mathematics, 3 (6).
Blanton, M., & Kaput, J. (in press).
Developing elementary teachers'
algebra "eyes and ears". Teaching
Children Mathematics.
Cai, J. (1998). Developing algebraic
elementary
in
reasoning
the
grades, Teaching Children Mathe-
matics, 5 (4), 225-229.
Carpenter, T. C., Franke, M. L.,
Thinking
(2003).
Levi,
L.
&
mathematically. Portsmouth, NH:
Heinemann.
Schifter, D. (1999). Reasoning about
operations: Early algebraic think-
ing, Grades K-6. In L.
stiff &
Mathematical
Curio,
(Eds.),
F.
1999 NCTM
reasoning,
K-12:
yearbook. (pp. 62-81). Reston, VA:
National Council of Teachers
of Mathematics.
Way. that Schoola
Can Support Change
Gamoran, A., Anderson, C. W.,
Secada, W. 0.,
A.,
Quiroz,
P.
Williams, T., Ashmann, S. (2003).
Transforming teaching in mathe-
matics and science: How schools and
districts can support change. New
Yorlc Teachers College Press.
Supporting professional development
and teaching for understanding.
(2002, Fall). in Bri. (Available at
http://www.wcer.wisc.edu/ncisla/
publications)
8 a 8
NATIONAL CENTER FOR IMPROVING STUDENT LEARNING & ACHIEVEMENT IN MATHEMATICS & SCIENCE
K-12 Mathematica & Science
POLICY CONSIDERATIONS
Algebraic Strategie6 tor Elementary Grade6
are evolving: (a) teacher communities, (b)
be applied in various contexts. For example,
n light of concerns about student perform-
I
Blanton and Kaput worked to integrate
principal communities, and (c) communities
ance on state, national, and international
MCAS items into the project as a way
reexamining
based on teacher-principal partnerships. The
educators
assessments,
are
to increase professional congruency. The
approaches to mathematics curricula and
researchers found that building connections
pressures across the district to perform well
among these communities allowed change to
teaching practices. The approach described in
this issue of in Brief suggests ways that
occur in a network of mutually supportive
on this state assessment made coverage of this
material relevant to the teachers. Thus, one
teachers can engage students in algebraic
relationships.
reasoning as they learn arithmetic. Impor-
of the tasks for teachers in the program was to
Linking leadership to professional devel-
algebrafy MCAS by identifying test items
tantly, the approach takes into account the
opment goals. The leadership academy, con-
that involved (or could be extended to
of teachers,
and constraints
capacities
ducted as part of the Massachusetts profes-
administrators, and available instructional
involve) algebraic reasoning and then include
sional development program, addressed areas
those items in their daily instruction.
materials. Described below are several ways
in which leaders could support teachers in
that administrators can support teachers in
Similarly, the Massachusetts district was able
integrating algebra into their practice. Activ-
this approach.
to connect the algebrafication approach
ities were patterned after those used in the
with the district's literacy initiative and,
teacher seminars
Supporting teacher learning and profes-
order provide
in
to
more recently, with another mathematics
participants with experience in algebraic
sional development. Administrative sup-
professional development program adopted
reasoning. (Teachers shared their classroom
port of content-rich professional develop-
experiences with principals and district
by the district. Rather than compete with the
ment that builds teachers' knowledge and
literacy initiative, which was long-running
leaders.) Specifically, the sessions focused on
their skills in developing algebraic problems
and had substantial funding, the team sought
and activities is a key part of transforming
e Understanding and supporting algebraic
ways to find synergies between the two
elementary mathematics teaching. Building
reasoning and practice so that the evalua-
programseach of which emphasized active
teachers' "algebra eyes and ears" requires an
tion and hiring of teachers reflected the
meaning-making and purposeful expression
initial focus on content knowledge and skills
needed move in this direction.
of ideas. Several teacher-leaders collaborated
that often means engaging expertise available
developing a year-long professional
e Assisting administrators in restructuring
in
in the district, state, or community. When
the school day to allow for ongoing
development agenda that would explore
teachers learn ways to transform arithmetic
algebraic reasoning in the context of the
teacher collaboration.
problems into algebraic reasoning problems,
learn how to
literacy initiative. As part of this collabora-
they also identify and
e Enabling teachers to promote mathemat-
tion, they looked for children's literature
opportunities integrating
for
organize
ics literacy on a school-wide basis and to
that could be used in conjunction with
algebraic reasoning in their classrooms, while
integrate it into other district initiatives.
existing algebraic tasks or in the creation of
their
own
simultaneously
developing
Common among the participating princpals
new tasks. They sought to connect initiatives
instructional resource base.
was their commitment to
in order to provide students additional ways
community
professional
Building
a
to access algebraic tasks and to increase the
e Work with teachers to enhance their
network. In the Massachusetts district,
potential frequency with which teachers
professional development.
NCISLA researchers Blanton and Kaput are
could integrate algebraic reasoning into
finding that a teacher community is critical
e Preserve teachers' autonomous role in
their classrooms.
in supporting teachers as they transform
and
professional
designing
leading
their practice. Developing a community
development.
takes time (often 3 to 4 years), and such a
Share decision-making authority with
community and its growth are sensitive to
that affected the
teachers on issues
changes in leadership and teacher attrition.
school community.
Establishing a network, however, can pre-
Developing congruency across educa-
vent isolation and support teachers as they
work to change their practice over time.
tional initiatives. Blanton and Kaput define
Blanton and Kaput propose establishing
developing professional congmeney as identify-
ing and strengthening ways in which teach-
what they describe as a professional commu-
nity network (Blanton & Kaput, in press-c),
ers' professional obligations and interests can
support each other (Blanton st Kaput, in
which would include several interconnected
but distinct communities that have parallel
press-c). Because the algebrafication strategy
purposes focused on a common goal. In the
is about transforming practice, not adhering
district described here, three communities
to a particular curriculum, the _approach can
I
NATIONAL CENTER FOR IMPROVING STUDENT LEARNING & ACHIEVEMENT IN MATHEMATICS de SCIENCE
K-12 Mathematica & Science
POLICY CONSIDERATIONS
Algebraic Strategie6 tor Elementary Grado
he research described in this issue of
T
Or, the teacher could vary the conditions
These principles guided teachers in developing
classroom tasks that
in Brief shows young students potential for
of the problem:
reasoning algebraically in conjunction with
s Involved sequences of computations
Assuming I make $2 per hour for babysitting, how
their learning of arithmetic. The professional
yielding numerical patterns that served to
many hours do I need to work to have enough
development approach taken by the research
engage students arithmetically.
money to buy the shirt that costs $14? $20? $P ?
team is described here in more detail to
What if I earned $3 per hour? How many hours do
provide educators a deeper look at the
s Promoted the use of non-executed num-
I need to work to buy the $14 shirt?
ways these teachers worked to change their
ber sentences (e.g., 0 + 1 + 2 + 3 + 4 +
+ 18
instructional practice.
+ 19) as objects for reasoning algebraically,
(Visit the NCISLA website at http://www.
rather than simply occasions to compute.
wcer.wisc.edu/ncisla, Teacher Resources, for
professional development
Establishing
sample Algebrafied Arithmetic Tasks.)
e Facilitated the algebraic use of number.
seminars. The cross-grade seminars provided
For example, a teacher might use so large a
elementary teachers both time and a place to
Developing classroom norms. A key compo-
number that arithmetic becomes an ineffi-
reflect on a common base of problems, which
nent involved in students learning to reason
they customized and solved individually.
cient way to solve the problem (e.g., 20 in
algebraically is active student involvement in
the third-graders' handshake problem),
As they progressed in their learning of the
proposing mathematical conjectures and
ways algebraic reasoning related to
and students are encouraged to think about
the
justifying their reasoning. Throughout the
teaching and learning of arithmetic, the
patterns and relationships (see page 1).
seminars, teachers were supported in finding
teachers tailored the problems to their own
ways to create a classroom culture that fostered
a Could be sequenced by a set of events or
grade levels, tried them with their students,
the kind of discussions needed to develop
objects, particularly through figural or
and then shared the results with their
students' algebraic reasoning skills.
physical enactment, such as shaking hands.
colleagues. Through the seminars, teachers
This provided a concrete starting point for
Detailed analysis of the case study data
gained insight into the ways problemsand
building patterns and relationships among
(Blanton & Kaput, 2002) revealed several
students' problem solving and reasoning
mathematical concepts.
of teaching practice
that
characteristics
strategiesdiffered across grade levels.
integrated algebraic reasoning and mathemat-
instructional
Transforming
materials.
Designing challenging problems. In the
In her classroom,
increased
"Jan's"
ics.
The seminars often began with the question,
cross-grade
seminars,
designed
teachers
sensitivity to algebraic reasoning meant she
"How can this one-numerical-answer problem
challenging mathematical problems that
was able to thread algebraic themes into her
be transformed into an algebraic-reasoning
provided students
with
in
experiences
conversations with students over sustained
problem that involves building and expressing
generalizing and formalizing patterns and
periods of time. Specifically, she was able to
a pattern or generalization?" This transforma-
relationships, as well as in justifying conjec-
tion was typically accomplished by varying
s Engage in a spontaneous and planned
tures. The problems were selected based on the
one of the numerical "givens" of the problem
algebraic treatment of number.
extent to which they
and then examining the patterns that emerged
e Integrate
into
algebraic
processes a
through developing a series of number sen-
s Addressed important mathematical ideas.
single task.
tences and calculations.
e Were approachable at different levels and
e Generalize
an
activity
introduce
to
For example, in the word problem
with different representations.
algebraic themes.
a Had potential to generate rich conversa-
I want to buy a tee shirt that costs $14 and have $8
Jan
in
the
students
engaged
algebraic
tions.
saved already. How much more money do I need to
treatment of number through
earn to buy the shirt?
e Involved substantial quantitative reasoning
s Discussion of number properties such as
and computation.
A teacher could vary the "givens" of a word
odd-even parity or commutativity.
problem to make it an algebraic task perhaps
Through collaboratively solving such prob-
e Use of symbols to represent unknown
by varying the cost of the item:
lems and redesigning tasks for students,
quantities.
the teachers and researchers outlined several
Suppose it cost $15, $16, $17, or $26. Using P for the
principles for designing algebraic reasoning
price of the item I want to buy, write a number
Variation of tasks along one dimension to
(Blanton & Kaput, in press-b).
sentence tkat describes how much more money
generate numerical patterns.
I need in order to buy the item.
8
1
I I
collaborate and participate in professional
development activities. The seminars also
provided principals and coordinators an
opportunity to discuss the approach with
In their
first
session,
teacher-leaders.
participants solved the handshake problem
before viewing a video of the third-grade
students solving the same problem. The
principals were impressed by their students'
apparently high capacity for mathematical
thinking, given the students prior perform-
ance on state assessments.
Professional development. Because this
mathematics
algebrafying
approach
to
most elementary
instruction
outside
is
teachers' experience, biweekly after-school
seminars focused on developing teachers'
algebra "eyes and ears." During each academic
year, approximately 50 teachers from 16
schools across Grades K-5 participated in sem-
inars that were led by teams of peer teacher-
leaders (usually two) who had undergone at
least one year of training with Kaput and
When Jan's Grade 3 students generated a number sentence for a group of 20 people shaking hands, the students
Blanton in similarly structured seminars. The
organized the numbers into 10 pairs, with the sum of each being 19. (See Focus onthe "Handshake Problem,"
researchers met monthly with these teacher-
page 4).
leaders to collect data and assess the effects of
the seminar activities on teacher practice.
Teacher Learning Through
activities considered to be the heart of algebraic
reasoning. For this reason, the professional
Protemional Development
Supporting Inatructional Change
development approach engaged teachers in
rich mathematical experiences embodying
vaput and Blanton's approach to professional
these activities.
The researchers identified the following
1N..development takes into account constraints
Rather than implementing a new instructional
factors that support the growth of a
on teachersnamely a prior orientation that
teacher community:
program and curriculum, this research and
focuses on arithmetic and computation without
professional development program focused on
8 Establishing grade-level school-based
integration of algebraic reasoning, as well as
enabling teachers to enhance their existing
study groups led by teacher-leaders.
textbooks and instructional materials that take a
mathematics resource base and incorporate
similar approach. At the same time, the
9 Engaging teacher-participants for a
instructional practices to promote students'
researchers recognize that teachers' capacity for
minimum of 1 year (2 or more years in
algebraic reasoning. Specifically, teachers were
mathematical and pedagogical growth can offset
most cases).
introduced to ways in which selected arithmetic
the challenges presented by those constraints.
9 Conducting seminars that emphasize
problems could be transformed into algebraic-
solving mathematical problems and
The professional development seminars, which
reasoning problems. By using teachers' at-hand
understanding students' thinking,
purposely involved teachers from across multi-
instructional materials as a base for this activity,
then using these as a catalyst for
ple grade levels, were structured around the
algebrafying became part of teachers' daily
thinking about teaching practice.
three activities discussed here. (More detail
practice. The process of finding and modifying
9 Using the resources teachers gener-
about these activities is provided in the Teacher
other problems helped build the skills needed to
ate as the basis for a shared, growing
Considerations insert.)
continue long-term growth and development
set of materials.
beyond the seminars.
Customizing and solving problems. The
9 Aligning the implementation of the
goal of this program was to make aspects of
Group discussion and teacher observations of
professional development program
their students' work allowed for comparison
algebraic reasoning (such as generalizing and
with other initiatives.
formalizing) part of mainstream instruction
across grade levels, giving teachers an opportu-
9 Integrating statewide assessment
nity to learn more about students' growth
rather than a form of occasional enrichment.
tasks into the teacher resource base.
little
have
Elementary
typically
trajectories and develop strategies for adjusting
teachers
activities accordingly. In later iterations of the
experience with generalizing and formalizing-
0
in Brief : ALGEBRAIC Stra egies for Elementary Grades
VoL.3, NO.1 : SUMMER 2003
seminar, locally produced teacher-tested activities
Focus onthe "Handshake Problem"
from one year became a resource for new teacher
participants in following years. The researchers
considered this important as a way to help
teachers see these materials as their own rather
The handshake problem engaged "Jan's" students in algebraic
Initially, the third-grade students acted out the
than as a set of externally provided materials to
reasoning.
multiple handshakes for smaller-sized groups and carefully
be "implemented."
recorded their data for different numbers of people.
The
Providing readings to support reflections.
research team provided readings that offered
When she gathered the students into a larger group, Jan led the
teachers an opportunity to reflect on the ways
students to generate a number sentence for a group of 20 people.
their approaches to mathematics instruction
The students came up with "19 + 18 + 17 + 16 +
+ 2 + 1 + 0".
differed from, or were similar to, other
At that point, Jan asked the students to consider a way to "change
instructional practices detailed in the education
the order of these numbers and make them easier to add up." The
research literature. (See the Resources for Teach-
students realized that they could organize the numbers into 10
ers and Policymakers listed on the bookmark in
pairs, with the sum of each being 19 (e.g., 19 + 0, 18 + 1, 17 + 2).
this publication.)
The following discussion ensued:
In the seminars,
Addressing classroom norms.
Now I've got to add up all these 19s. What is this?
JAN :
teachers considered ways to foster classroom
STUDENT: Repeated addition. You could do times.
that
students'
would
discussions develop
algebraic reasoning skills. The processes of build-
I could do times?
JAN :
ing patterns, making conjectures, and generaliz-
STUDENT: 19 times 10 lpairsl-190.
ing and justifying mathematical facts and
relationships are at the core of the algebraic
How did you figure that out so quickly?
JAN :
reasoning strategy. Teachers focused on these
STUDENT: I just changed that to 9 and added a zero.
issues and considered questions such as whether
a culture of inquiry was developing in their class-
Why?
JAN :
rooms, what the classroom norms for argumenta-
The student explained that he had made the "1 into a 100 and 9
tion were, and whether students questioned each
into 90" by using 10 times 10 to get 100, and 9 times 10 to get 90,
other and expected justification of mathematical
then added.
statements. Teachers also discussed the extent to
which their students were learning computation
Through the handshake problem, Jan's students came to
and justification skills, the differences in instruc-
recognize a pattern that led to an important generalization
tional approaches and experiences, and the
about how to calculate the sum of any arithmetic series.
In
difficulties involved in changing practice.
solving a particular problem, they generated and justified a
procedure that can be applied to find the sum of any
Next Stepa
arithmetic series.
The separation of algebra from early mathemat-
To View a Video of Students
ics instruction dates back to the 1400s (Swetz,
Solving the "Handshake Problem"
1987), but this separation is not appropriate
contemporary mathematics
instruction.
for
Researchers suggest that, when introduced in
Visit Annenberg/CPB at http://www.learner.org/catalog,
elementary grades, algebraic reasoning paves the
or call 7-800-LEARNER and ask about
way for more advanced learning in middle and
the "Looking at Learning... Again, Part 2" video series.
high school and strengthens student understanding
of arithmetic. This issue of in Britf offers a glimpse
This video is also featured in a forthcoming NCISLA multimedia
into a content-rich professional development pro-
Powerful Practices in Mathematics and Science:
package
gram that develops teachers' "algebra eyes and ears."
Modeling, Generalizing, and Justifying
to be released
When considering long-term improvement plans,
through the North Central Eisenhower Mathematics and Science
schools and districts should consider allocating
Consortium at the North Central Educational
Laboratory.
professional development resources toward similar
approaches, which do not necessarily require the
To order Powerful Practices, e-mail [email protected].
purchase of new curricula or instructional pro-
grams (see insert on Policy Considerations).