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~r.e
Err
The emphasis on "teaching the whole child and the whole curriculum" advocated
by some can be seen as opposed to the emphasis on "teaching mathematics content
area" as advocated by others.
MATHEMATICS IN THE MIDDLE: BUILDING A FIRM FOUNDATION OF
UNDERSTANDING FOR THE FUTURE
Glenda Lappan, University Distinguished Professor, Department of
Mathematics, Michigan State University.
THE MIDDLE SCHOOL LEARNER: CONTEXTS, CONCEPTS, AND THE
TEACHING CONNECTION
Thomas Dickinson, Professor of Curriculum and Instruction at Indiana State
University.
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Glenda Lappan
University Distinguished Professor, Department of Mathematics,
Michigan State University
What makes mathematics in the
middle grades so important? There are
factors that have to do with adolescent
growth and development. Others have
to do with the subject matter itself, its
increasing complexity, and its increas-
ing importance as a foundation on
which science, mathematics, and tech-
nology literacy is to be built. Each of
these factors that influence mathemat-
ics' importance in the middle years will
be elaborated individually and as they
interact with each other.
Students in the middle grades go
from pre-adolescence to adolescence at
different rates. At the beginning of the
year in grade six, students may all look
much as they did the year before in
grade five. By the end of this year or
the next, most students, both girls and
boys,willgrowseveralinches. Stu
1 , ~, ~
. . .
dents reactions to these rapid changes
in their bodies are, of course, quite
varied. But for all students these
changes are accompanied by emotional
challenges. Students are faced with
getting to know a new self, including a
new body, with new emotions. Strong
,' ,. .
- .
among these new emotions is a need to
be like others, to belong. Parents are no
longer the center of the world. Peers
are now the focus for what is "cool" and
~ . . .
tor emotional support or crushing
· . · ~ . ..
rejection. At the same time, a student's
intellectual capacity to reason is expand-
ing rapidly.
Students in middle grades are grow-
ing in their ability to reason abstractly.
They become capable of generalization,
abstraction, and argument in math-
ematics. This signals the need for
programs that give students the oppor-
tunity to expand their experiences with
"doing" mathematics, with controlling
variables and examining the conse-
quences, with experimenting, making
conjectures, and developing convincing
arguments to support or disconfirm a
conjecture. Taken together, these
changing intellectual capacities and the
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fragile emotional state of these vuIner
able middle grade students call for very
carefully crafted mathematical experi
ences that allow students to satisfy
their social needs as well as their
intellectual needs. The following
section looks at the mathematics of the
middle grades, its challenges and its fit
with these emerging adolescents.
MATHEMATICS OF THE
MIDDLE GRADES
As students encounter mathematics
in the upper elementary gra(les, the
emphasis changes from a focus on the
additive structure of numbers and
relationships to the multiplicative
structure of numbers and relationships.
This means that the students are faced
with new kinds of numbers, fractions operations
and decimals that rely on multiplication
for their underlying structure. These numbers
numbers are useful in making new
kinds of comparisons that rely on two
measures (or more) of a phenomena.
For example, which is best, 5 cans of
tomatoes for $6 or ~ cans for $9? A
simple subtraction will not resolve the
issue. A comparison that takes into
account both the quantity of cans and
the prices is called for.
These new mathematical ideas contain
intellectual challenges for the students
that are as conceptually difficult as any
thing anywhere else in the K-12 math
.
T E A C H I N G A N D L E A R N I N G MAT H E MAT I C S
emetics curriculum. Many ofthese
mathematical ideas win not reach their fuD
maturity in the mi(l(lle years, but it is in
the mi(l(lle gra(les that the firm foundation
for understanding is laid. It is here that
students have time to experiment, to
ponder, to play with mathematical ideas,
to seek relationships among ideas and
concepts, and to experience the power of
mathematics to tackle problem situations
that can be mathematized or modeled. It
is also here in the middle years that the
serious development of the language of
mathematics begins.
It is easy to make a laundry list of
topics in mathematics important in the
middle years. Here is such a list:
· Number theory, factors, multiples,
division, products, relationships
among numbers and among
· Rational numbers, integers, irrational
· Fractions, decimals and percents
· Ratios, rates, proportions, quantitative
· , ~
reasoning aria comparison
· Variables, variables changing in
relation to each other, rates of change
among related variables
· Representations of related variables in
tables, graphs, and symbolic form
· Slope, linearity, non-linear relation-
ships, families of functions
· How things grow in both an algebraic
and a geometric sense
Maximum and minimum
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· Shape and shape relations, similarity,
symmetry, transformations
· The Pythagorean Theorem
Chance and reasoning about uncertainty
Prediction using probability and
statistics
· Measurement systems and measure-
ment of attributes such as length,
area, volume, weight, mass, angle,
time, distance, speed
· Visualization and location of objects in
.
space
Representation of three-dimensional
objects in two-dimensional drawings
and vice versa
The real challenge is to help students
see these ideas in their relationship to
each other. This requires a different way
of bringing mathematics and students
together. Rather than see these ideas as
a series of events to be covered, good
middle school curricula are finding ways
to help students engage in making sense
of these ideas in their complex relation-
ship to each other. Symbolic representa-
tions of patterns in algebra are seen as
related to symbolic representations of
transformations, slides, nips and turns, in
the plane. In other words, we need to
think about how we set students' goals
for mathematics. Rather than expound-
ing a list of individual ideas to be "cov
ered," goals need to help a teacher and a
student see what is to be learned, how
the ideas can be used, and to what the
ideas are connected.
What follows is an argument orga-
nized around strands of mathematics
and important related i(leas within those
strands for thinking about what is
important for students to learn in
mathematics in the middle years. In
each of these sections an argument for
inclusion is based on the universality of
the i(leas in the strand no matter what
the future ambitions of the students will
be. Whether students will enter post-
secondary educational institutions or
the world of work directly after high
school, the following ideas are key.
They are mathematics that is central to
being able to manage affairs as an adult
and to be a good citizen who makes
good decisions based on evidence
rasher then persuasive rhetoric. They
are also mathematical ideas that have
their roots in the middle years.
RATIONAL NUMBERS AND
PROPORTIONAL REASONING
One area of mathematics that is
fundamental to the middle grades is
rational numbers and proportional
iThis strand argument is taken from a paper I presented at the Fourth International Mathematics
Education Conference at the University of Chicago in August 1998 entitled Preparing Students for
College and the Work Place: Can We Do Both?
MATH EMATI C S I N TH E Ml D DLE
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reasoning. This includes fractions,
decimals, percents, ratios, rates, propor-
tions, and linearity, as well as geometric
situations such as scales and scaling,
similarity, scale factors, scale modeling,
map reading, etc. Another important
aspect of this mathematics is that it
gives a powerful way of making com-
parisons. Having two measures, rather
than one, on attributes that we are
trying to compare leads us into the
world of derived measures that are often
per quantities or rates. This is a core of
ideas that relate to quantitative reason-
ing or literacy and connects directly to
science. Almost everything ofinterest
to scientists is a quantity, a number with
a label. Study of mathematics involving
quantitative reasoning invariably means
reasoning about mathematics in con-
texts. Part of what makes mathematics
so powerful is its science of abstraction
from real contexts. To quote Lynn
Steen (p. xxiii, 19971:
The role of context in mathematics poses
a dilemma, which is both philosophical
and pedagogical. In mathematics itself....
context obscures structure, yet when
mathematics connects with the world,
context provides meaning. Even though
mathematics embeciclec3 in context often
loses the very characteristics of abstrac-
tion and clecluction that make it useful,
when taught without relevant context it is
ad but unintelligible to most students.
Even the best students have difficulty
applying context-free mathematics to
T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S
problems arising in realistic situations, or
applying what they have learned in
another context to a new situation.
This is an argument that works for
students hea(le(1 for the workplace as
well as college. Clearly applications, but
even more so, mathematics skills, pro-
cedures, concepts, and ways of thinking
and reasoning taught through contextu-
al~zed problem situations, help students
of all kinds and ambitions learn math-
ematics. The modern high-performance
workplace involves problems that require
sophisticate(1 reasoning anti yet often
only the mathematics of a good middle
school education. The high level of
reasoning that empowers a student to
use these ideas in new and creative ways
is not complete by the end of middle
school but requires continue(1 experi-
ences in the high school curriculum.
DATA ANALYS15,
REPRESENTATION, AND
INTERPRETATION
Another aspect of mathematics that
pervades our modern life is (lata, or as
statistician David Moore cabs it, numbers
with a context. In the workplace, in jobs
at all levels, employees are (leafing with
the problem of either reducing or un(ler-
stan(ling the reduction of large quantities
of complex (lata to a few numbers or
graphical representation. These num
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hers or statistics are expressed in frac-
tions, decimals and percents. Thus,
knowing something about data analysis
and interpretation and rational numbers
is important to ad students. The society
in which we live has become quantitative.
We are surrounded by data.
To han(lle our money, to make trans-
actions, to run our businesses, as well
as to do almost any job well, we need
mathematics that is underpinned with a
high level of reasoning and understand-
ing. As Iddo Gal puts it, "There are no
"word problems" in real life. Adults face
quantitative tasks in multiple situations
whose contexts require seamless
integration of numeracy and literacy
skills. Such integration is rarely dealt
with in school curricula" (Gal, 1997,
p. 361. But, it should be! Using con-
texts that show the world of work help
make mathematics more authentic to
students regardless of their future
career goals. Dataanalysis,representa-
tion, and interpretation are an important
and rich area for such examples.
GEOMETRIC SHAPES, LOCATION,
AND SPATIAL VISUALIZATION
Whether we are talking about an
intending calculus student or an employee
at a too} and die shop, geometry, location,
and spatial visualization are important.
Just as a calculus student nee(ls to visual
MATH EMATI C S I N TH E Ml D DLE
Size a curve moving to form the boundary
or surface of a solid in space, a too} and
die employee uses such skis to set a
machine to locate and cut a huge piece of
steel to specifications that truly exceed
our imagination. Here again is an area of
mathematics that has application in the
workplace and should be a part of the
education of all students. This is also a
part of mathematics in which strand or
areas intersect. While ~ have labeled this
area geometric shapes, location, and
spatial visualization, algebra becomes a
too} for specii~ring both location and
shape. Even the spatial visualization
aspects of geometry are enhanced by
experiences with transformational geom-
etry and its related algebra. This is also
an area with connections to measurement
which means rational numbers are
essential. We cannot talk about any aspect
of measurement without being able to
deal with a continuous number line where
rational numbers and irrational numbers
fib in the spaces between the whole
numbers. So again we see connections
among strands of mathematics. Another
area that has much to offer to both the
world of work and to higher education is
chance.
CHANCE
Everyday humans are faced with
reasoning under uncertainly. Many
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adults make wrong decisions as a result
of not understanding the difference
between random variables, correlated
variables, and variables that influence or
cause certain behavior in another
variable. Probability is an area of
mathematics that is important across
the board. Itisa very challenging
subject and involves a practical every-
day kind of mathematics and reasoning.
Here, as in other strands we have
discussed, the rational numbers play an
important role. Probabilities are num-
bers between O and ~ that express the
degree of likelihood of an event happen-
ing. In experimental situations, prob-
abilities can be estimated as relative
frequencies again a use of fractions
and proportional reasoning. In other
situations, probabilities can be found by
analyzing a situation using an area of ~
square unit to represent everything that
can happen. This makes a connection to
geometry and measurement.
So much of the information with which
we are surroun(le(1 in our modern worI
is of a statistical or probabilistic nature.
Here the conception of mathematics as a
discipline that comes as close to truth as
we know it in the modern world and the
idea that what mathematics predicts may
not in fact happen in every instance
clash. Students learn to deal with the
notion of random variables anti the
seeming contradiction that there can be a
science of predicting what happens in the
T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S
long run when what happens in an
instance is random and unpredictable.
These are hard ideas that take a great
deal of experiencing before they seem
sensible and useful. ~ would argue that
the study of chance or probability should
be a part of the curriculum for all stu-
dents in the middle grades. It is in these
years that students have the interest and
we can provide the time to experiment
with random variables. Students need to
collect large amounts of data both by
hand and with technology simulations, to
see that data can be organized to build
models that help predict what happens in
the long run, an(1 our own theoretical
analysis of situations can help us build
predictive probability mo(lels. Now an
argument for one last area algebra.
ALGEBRA AND ALGEBRAIC
REASONING
~ think it is pretty clear that few
workers meet a quadratic equation to
solve on a (laity basis, or for that matter
few ever meet higher (legree polynomial
equations to solve. But they do deal
with formulas on a regular basis that are
quite complex. Figuring the exact
amount of a chemical cocktail to inject
into a cancer patient (leman(ls complex
measures anti uses of formulas. The
consequences of a mistake raise the
stakes a great (leal. There are so many
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examples of such in the world of work
that a strong case can be made that all
students do need algebra, both from a
functions point of view and a structure
or equation solving point of view. In fact
algebra that integrates with other areas
of mathematics and that uses applica-
tions as a driving force can reach all
students no matter what their career
goals. For the future mathematician or
scientist, algebra is fundamental. For
the future office worker, algebraic
thinking can mean the difference
between remaining in a low-level job
anti advancing to much higher-paying,
more demanding jobs.
~ do have to express one caveat. If
this is the dry memorized algebra of the
past, then it is as likely to damage
middle school students as help them.
However, an algebra strand that at some
grade levels has a real concentration on
algebra and algebraic thinking can
benefit all students. This means that the
ideas wait develop over time. This
means that applications wait drive the
learning of algebra, not just be the
problems that you try after you memo-
rize the types of problems in the section
of the book. This means that the funda-
mental ideas of algebra wall be taught in
such a way that students have a real
chance of (leveloping (jeep un(lerstan(l-
ing as they develop technical profi-
ciency. It should also be an algebra that
takes full advantage of calculators and
MATH EMATI C S I N TH E Ml D DLE
computers as tools to explore ideas and
to carry out procedures.
Rather than belabor my point, let me
just make it outright. We need to
develop a set of mathematical expecta-
tions (luring the mi(l(lle years that are
for all students. At some stage of high
school, students can benefit from
options of continuing in a statistics,
discrete mathematics direction or
toward calculus and more formal math
emetics. These should be real choices.
equally valued and equally valuable
depending on the post-secondary
directions a student aims to take. But in
the middle years, we need to develop
curricula that preserve student's options
for those future choices. However, such
a curricula alone will not make an
excellent mathematics experience for all
students. Teachers and teaching matter.
What teachers do with the curricula,
what the expectations are for students,
how students are expected to work,
what conversations are expected and
how these are conducted matter. Teach
ing in the middle grades requires
knowledge of the subject matter and
also of students at this age.
TEACHING MATHEMATICS IN
THE MIDDLE GRADES
What are the challenges associated
with teaching mathematics in the
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middle grades? Not all students reach
the same levels of cognitive growth at
the same time. This means that the
middle school mathematics classroom
teacher has the challenge of creating an
environment that supports students'
mathematics growth at many different
levels. Since middle school students are
at many stages of cognitive, physical,
and social development, the teacher
needs to understand where students are
in their mathematical growth. Typical
paper and pencil computation driven
tests give little specific understanding of
student's thinking. Thus, an additional
challenge for the teacher is creating and
using many opportunities for assessing
student understanding. Teachers are
experimenting with many new forms of
assessments varying from partner or
group tests or performance tasks to
projects or student portfolios.
Middle grades students have many
social, emotional, and physical chal-
lenges with which to deal. In order to
capture their attention and direct their
growing cognitive powers on mathemat-
ics, the mathematical tasks posed for
students must be focused on things they
find interesting and important. Things
that we as adults think of interest to
middle school students are not always
on target. This argues for opportunities
for choice within the mathematics
classroom. Choice of project topics,
choice of problems of the week to work
on, and even dealing with problems that
T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S
allow different strategies for solution,
are ways of allowing students to fee} the
power of choice. In addition, the kinds
of problem settings used for tasks need
to take into account the interests of
students at this level.
In or(ler for students to make sense of
concepts, ways of reasoning, productive
procedures, and problem solving strate-
gies in mathematics, and to develop skill
with number and symbolic operations
and procedures, they must be engaged in
exploring, investigating, inventing,
generalizing, abstracting, and construct-
ing arguments to support their ideas and
conjectures. New mathematics materials
for the middle grades are problem
oriented. Interesting contexts for investi-
gation are posed. Within these contexts
students bump into mathematics that
they need but do not yet have in their
tool kit. This need drives students to
invent, to create strategies for solving
their problems. Out of these student
ideas come the conversations that, with
the teacher's guidance, help make the
underlying mathematical ideas, concepts,
procedures, skills, and arguments more
explicit for the students. As you think of
the socialization of the middle school
level, there is a natural match between
the mathematical need to deeply under-
stand ideas and the social need to inter
act with ones peers around tasks that are
of interest. We can use this energy and
need to socialize to advantage in math-
ematics by refocusing our instruction to
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be more on mathematics as an experi-
mental science where seeking to under-
stand why is the major quest. Another
too} that we have available to us in
teaching mathematics is technology.
Technology is engaging to students. It
also allows access to mathematics that
could not be explored in the past. It
helps to create environments in which
students can see change, can engage in a
dynamic way with mathematical conjec-
turing. It allows students to tackle real
problems with messy data. It allows
students control over different forms of
representation of mathematical relation-
ships. It allows students to invent new
strategies for tackling problems; it allows
students to reason through problems in
ways that are different from the strate-
gies we would use. Again this freedom
to experiment and to create makes
mathematics of greater interest to many
more students. It connect mathematics
to the real world and to the interests of
these emerging adolescents.
SUMMARY
While mathematics in the middle
years is cognitively demanding, it is
suited to the developing capabilities of
MATH EMATI C S I N TH E Ml D DLE
middle school students. New forms of
classroom interaction, more emphasis
on experimentation, on larger, more
challenging problems, on seeking to
understand and make sense of math-
ematics and mathematical situations, on
using technology as a too} these are
the ways in which new middle school
curricula are seeking to engage these
social, energetic, (leveloping a(loles-
cents in learning for their own future.
Mathematics can be a key to opening
doors or to closing them for students.
Our goals has to be to help preserve
future options for our students until they
reach a level of maturity to understand
the consequences of their decisions. We
cannot do this unless we can make
mathematics interesting and relevant to
our students.
REFERENCES
Gal, I. (1997~. Numeracy: Imperatives of a
forgotten goal. In L.A Steen (Ed.), Hey
numbers count: Quantitative literacyfor
tomorrow's America (pp. 3~44). New York:
College Entrance Examination Board.
Steen, LA (1997). Preface: lithe new literacy.
In LA Steen (Ed.), Hey numbers count:
Quantitative literacyfor tomorrow's America
(pp.xv-xxvii0. New York: College Entrance
Examination Board.
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Thomas Dickinson
Professor of Curriculum and Instruction, Indiana State University
SMALL STORIES'
Successful middle schools have small
stories "because the school has time for
small incidents before they become
large" (Lipsitz, 19841. ~ have one such
story to share.
It was one of those early spring days
that promised clear skies and warm
weather for weeks to come. School
had just let out, and ~ was sitting at my
(leek, gra(ling papers. Wrappe(1 up in
the process of marking and recording,
~ didn't immediately see the small
figure at my door. "Mr. Dickinson, are
you busy?" the voice asked and my
grading reverie was broken. It was
Amy Bounds, a seventh grader in my
homeroom and a student in one of my
social studies classes. Amy was one of
those students that make up the bulk
of our classes, best described by
phrases such as "good student" or
"neat kill," one of those in(livi(luals
that we honor with the label of "nice to
have in class."
The question repeate(1 itself, "Mr.
Dickinson, are you busy?" ~ shook my
hea(1 anti wave(1 her on into the room
where she took the seat next to my
desk. 'what are you doing here Amy?
You shoul(1 be outside on a (lay like
this." ~ tol(1 her. Her reply abruptly
change(1 the (letache(1 moo(1 ~ was in.
"Mr. Dickinson, I've got a problem."
~ attempted to fight off my immediate
a(lult questions what kin(1 of problem
coul(1 Amy have? An(1 as my min(1 race
with possibilities, it (li(ln't slow (town
with her next comment. 'Nou're a man,
Mr. Dickinson, so you'll understand my
problem." Gathering my strength, ~
i"Small Stories" was originally published, in an expanded version, in the November 1988 issue of
Middle Schoolloavrnal. Used with permission.
OCR for page 33
asked Amy, 'VVhat's the problem? How
can ~ help?"
'VVell," she sai(l, "where (lo you get
your ties?"
"My ties?" ~ asked, trying to put two an
two together, unsuccessfully. "My ties?"
'yeah, your ties. My father's birthday
is tomorrow and I'm not going to let my
mother buy my present for him. ~ want
to buy him a tie like the ones you wear.
And I've been saving my lunch money.
Do you think ~ have enough?"
And with that Amy extracted from her
backpack a plastic bag stuffed with
change anti ploppe(1 it on my (leek.
Amy Boun(ls was growing up. She
was, to use her phrase, "no longer a
baby." And this translated most immedi-
ately into a problem with her father's
birthday. She wanted to buy her father
a present, an appropriate present, anti
she wanted to do it herself. But who
could she talk to, who could she ask?
Not her father. It was his birthday, and
she wanted very much to surprise and
impress trim. Not her mother. She
might not understand, or she might try
to interfere. So ~ was chosen, partially
because ~ was her homeroom teacher,
partially because ~ was a man, and
mostly because she liked the ties ~ wore.
Before we did anything we sorted
the money and talked about her father.
~ asked about his taste in clothes,
colors, and fabrics. Amy could answer
all my questions because she'd
THE MIDDLE SCHOOL LEARNER
checked his closet, several times.
During our talk Amy related that she
didn't have a regular allowance, that
she had been saving "extra" from her
lunch money for the last three months.
She'd also been lugging it around in
her backpack since she didn't want to
risk it being discovered at home. She
also knew better than to trust her hall
locker. And once a day she'd been
han(ling it over to her best frien(1 for
safekeeping while she took gym. Most
interesting to me was Amy's revelation
that she'd been counting it every night,
hoping that she'd have enough before
her father's birthday.
We talked ties for quite a while rep
stripes, paisleys, solids, Italian silks, and
knits. She ha(1 a list of the ones ~ wore
that she like(l, anti we went through her
list and discussed all the possibilities.
Finally Amy decided on something silk,
something blue or blue/gray, anti
something with a small conservative
pattern. And then she asked her origi-
nal question again, ' Where (lo you get
your ties?" ~ ha(1 been buying clothes
from one men's store ever since I'd had
my own money. ~ drew a map for Amy
of where the store was located. Would
she be able to fin(1 it? She no(l(le(1 with
a quiet "yes." Did she want the present
gift wrapped? They would (lo it for free,
but she ha(1 to ask. Did she have a car
already? No, well, there was a card
shop two doors down.
OCR for page 34
And then everything was settled and
in place. She left with a grin and a quick
thank you. After she was gone ~ went to
the office to make a telephone call. The
next day ~ had a note on my desk at the
start of homeroom "Everything is set.
Thanks." That evening ~ received a
telephone call from Amy's mother.
Years later ~ only remember her 'Thank
you." What ~ do remember as if it were
yesterday was the grin on Amy's face as
she came into homeroom the next day
and her comment, "He wore it this
morning, and it looked terrific!"
DEVELOPMENT IN CONTEXT:
THE CHILD, THE SCHOOL,
AND SOCIETY
You teach Amy. Or you teach teach-
ers about Amy. Or Amy may sit across
from you at your dinner table. Amy is
an individual. She is a human being.
She is a person. She is not a list of
developmental characteristics. She is
not a "typical" young adolescent (there
aren't any). She is instead, Amy. And
that is enough, if we would but see and
acknowledge it.
~ began with a story about a young
a(lolescent not in a mathematics
context on purpose. That purpose was
so that all of us would see development
for what it is: a natural anti normal
process, boun(le(1 by general psycho
T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S
logical guidelines, but embodied in
individuals. And this embodiment is
bounded, shaped, and influenced by a
range of contexts the individual
context of self, that mixture of heredity
and randomness; the context of family,
street, neighborhood; the larger social
context, which today includes an over-
whelming and disturbing mass media
context that impacts young adolescents
at their every turn. (Half of all young
adolescents spend three hours or more
each day watching television.) The
average young a(lolescent is expose(1 to
the mass media a total of about four
hours per (lay. Only sleeping anti
attending school occupies more time. ~
won't even go into the content if you'd
like, watch MTV for an hour, or flip
through the latest issue of any popular
magazine.
To try to un(lerstan(1 (1evelopment for
young adolescents we have to frame
these contexts with their in(livi(luality.
To put it bluntly, we have to understand
the in(livi(lual to un(lerstan(1 their
(levelopment. Goo(1 mi(l(lle schools try
to (lo just this. They try to un(lerstan(1
anti teach in(livi(lual children, not types
or categories of children, but in(livi(lual
chil(lren themselves. This is what This
We Believe: Developmentally Responsive
Middle sieve! Schools (National Mi(l(lle
School Associations, 1995) talks about.
It's what Turning Points: Preparing
American Youth for the 21st Century
OCR for page 35
(Carnegie Council on Adolecscent
Development, 1989) recommends.
And it's what I've been seeing in a small
school in San Antonio, Texas.
DEVELOPMENT AND THE
TEACHING CONNECTION
Donna Owen looks out over the
overhead projector at a sea of khaki and
white and asks "What's my rule?" and
points to the bright yellow poster on the
board. Two columns of numbers lie
waiting for her young charges:
IN | OUT |
3 11
5 15
o 5
10 25
The students are already at work on
this daily ritual. These are 6th graders,
in uniforms (that's the khaki and white)
at the American Heritage School, an
Outward Bound Expeditionary Learning
(ELOB) school-w~thin-a-schoo} that is
part of Edgar Allen Poe Middle School in
San Antonio, Texas. The vast majority of
students are Hispanic. Their school, a
new building, stands in stark contrast to
its neighborhood surroundings.
Kids, seated in pairs, are tackling the
problem some with charts of their
THE MIDDLE SCHOOL LEARNER
own, others with series of problems
trying to arrive at a formula that works
for all of the four pairs, one student
using a number line. Around the room
there is a rich riot of color number
lines, bought and teacher-made posters
with vocabulary lists anti mathematics
operations, and two computers (still not
hooked up to the internet it's Novem-
ber when this is being written) in the
back of the room by the teacher's desk.
The kids continue to work focused, on
task, quiet.
What first drew me to this teacher is
outside. On one side of the hall is a
large gallery (think about that wor(1 as
oppose(1 to a "bulletin boar(l"), maybe
15 feet long. On it are student charts
and graphs. But not just charts and
graphs, but charts anti graphs about
the students. In preparation for the
team's first "expedition" where the
topic was the Americas, stu(lie(1
through the lens of the focus question
"How is the good life in the Americas
nurture(1 anti challenge(l?" Donna had
her sixth graders work on charts and
graphs by collecting data on them-
selves. Working in pairs the students
queried their teammates, recorded the
data and graphed or charted it, and
then wrote narrative explanations of
their findings. There were bar graphs,
line graphs, pie charts. And the topics
of the charts and graphs were wide
ranging:
OCR for page 36
· the males and females in the class
· the numbers of letters in students'
names
· the days of the week students were
born
· the right- and left-handedness of
students
· the number of lima beans that each
student coul(1 pick up with one han(1 (!)
· the height of individuals in the class,
in centimeters
· the circumference of individuals'
heads, also in centimeters
And all this work was displayed (and
gra(le(1 too). The accompanying narra
tives demonstrated:
· a sense of audience (especially in
their detail and completeness)
· a concern for language, structure,
and correctness
· an appreciation of self and others
(peers) as a source of information
that could be used in mathematical
understanding
Donna Owen was teaching individu
als. She was also teaching mathematics.
And she was also helping students
understand themselves. She was
placing (levelopment within a context of
in(livi(luals anti those in(livi(luals within
a rich and involving context of learning,
in this case mathematical un(lerstan(l
ing. And if you're wondering about this
marvelously insightful teacher, here's
the bio: she is late to teaching; at 29 she
T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S
went off to get a degree after serving as
a teacher's aide in a classroom where
she was getting more and more respon-
sibility because of her talent, insight,
and drive. Encouraged by her teacher
anti principal to pursue a (1egree anti
teaching license, she (li(1 just that anti
became the first of seven children (she's
the ol(lest) to go to college. An(1 she (li
her degree in three years, cum laude,
taking 18-21 hours of course work at a
time, arranged on three days a week so
that she could continue to do her aide
work. By the way,Donna has taught
middle school English and journalism
for the past ten years. She thought she
needed another challenge so she got
licensed in mathematics and is in her
secon(1 year of teaching that subject.
Last year, her first teaching mathemat-
ics, 91% of her students passed the
Texas Assessment of Academic Skills.
I'm back in the classroom again. The
secon(1 expedition is un(ler way. Stu-
(lents are working on problems involving
space travel. Along one entire was,
probably 40 feet, is a swath of purple
paper, door to ceiling, with the sun at one
en(1 anti off on the other si(le of the room
(actually on the back of the door), Pluto.
This is the entire solar system, to the
scale of 1" = 10,000,000 miles. An(1 the
sizes of the planets are to scale as well.
The students are working on estimated
travel time in hours to the planets. It's
another example of 'VVhat's my rule"
OCR for page 37
where He kids are denv~ng the rule Tom
data. ~ watch, amazed at Be degree of
engagement. At Me end of We class my
notes summarize the lesson:
· focus on reading directions
· finding necessary information
· using available data
· drawing conclusions Tom data
working in cooperative groups
recording information accurately
· doing basic mathematical operations
.
working alone
· making choices to investigate
individually
~ could tell you all the developmental
psychology connections here, but ~ hope
you see them. ~ know my readers,
THE MIDDLE SCHOOL LEARNER
being mathematically skilled, can see
what's happening here mathematically.
This is another "small story." And like
most small stories in good middle
schools it goes relatively unnoticed,
except by the teacher and students who
are growing and changing together, as
individuals.
REFERENCES
Carnegie Council on Adolescent Development.
(1989~. Tavrning Points: PreparingAmerican
Youth for the 21st Century. New York: Carnegie
Corporation.
Lipsitz, J.S. (1984~. Successful schools foryoa`ng
adolescents. New Brunswick, New Jersey:
Transaction Books.
National Middle School Association. (1995~. This
we believe: Developmentally responsive middle
level schools. Columbus, OH: Author.
OCR for page 38
Representative terms from entire chapter:
middle grades
