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OCR for page 39

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Art
The sessions on content and learning mathematics in the middle grades focused
on the questions:
.
What developmental considerations are important in thinking about middle school
students as learners? As learners of mathematics? Are these compatible?
· What do we know about middle school students' capacity for learning? For learn-
ing mathematics?
· What are important ideas in mathematics for the middle grades and how are these
related to developmental learning considerations?
REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS
Nancy Doda, Professor of Education, National-Louis University.
MATHEMATICS CONTENT AND LEARNING ISSUES IN MIDDLE GRADES
MATHEMATICS
Kathleen Hart, Chair of Mathematics Education (retired), University of
Nottingham, United Kingdom.
SUMMARY OF SMALL GROUP DISCUSSION ON CONTENT AND
LEARNING ISSUES IN MIDDLE GRADES MATHEMATICS

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Hi
',,r~.r~--~
Nancy Doda
Professor of Education, National-Louis University, Washington, DC
OVERVIEW
Young adolescence is a remarkable
and challenging time in human life not
always appropriately appreciate(1 nor
well understood. In 1971, loan Lipsitz
published a review of research on the
middle grades child and learner in
which her chosen book title, Growing up
Forgotten, was essentially her most
provocative conclusion that young
adolescents were American education's
most neglected and least well under-
stoo(1 age group (Lipsitz, 19711. Since
this seminal work's debut, there has
been without question a steady ca-
cophony of hearts and minds that have
joined to recognize, celebrate, and
better understand and serve this unique
age group. Indeed, several decades of
reform initiatives now stand before us,
yielding wisdom to guide future efforts
to improve schooling for young a(loles-
cents (Lipsitz, 1981; George and
Shewey, 19941.
While we are fortunate to have this
rich history of mi(l(lle gra(les reform,
with its now well-documented dividends
(e.g., FeIner et al., 1997), it is nonethe-
less clear that the (livi(len(ls of greater
student learning and achievement are
still more illusive than we might have
hoped. There still remain enormously
stubborn achievement gaps between
white children and children of color;
between poor children and financially
advantaged children, between girls and
boys. If asked the question, "How are
we doing in the U.S. with regards to
student achievement?", the truthful
answer would be, 'That depends on
which children we are discussing." And
in the case of our focus here, perhaps it
also depends on which subject we are
· ~
examining.
We face a crossroads in middle school
reform. One that calls upon us to
reexamine not only the fundamental
philosophy of the middle school con-
cept, but the beliefs and practices that

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remain as potential deterrents to our
gravest challenge that of ensuring
academic success for all our children.
There is an urgent need to (lig (1eeply to
understand what is needed in the
reform conundrum, particularly with
student learning in mathematics. Stu-
dent success in middle school math-
ematics remains the great equalizer or
divider, as it were, and our students'
individual and our collective futures
depend upon it.
CHANGING THE IMAGE-
CHANGING THE CURRICULUM
While the contemporary dialogue
about the nature of young adolescents
has often been delivered with affection-
ate humor, such humor often highlights
the least dignifying portrait of the young
teen. ~ have often quoted Linda Reiff's
comments to evoke an affirming
chuckle from most middle grades
educators or parents. She wrote:
Working with teenagers is not easy.
It takes patience, humor and love. Yes,
love of kids who burp and fart their way
through eighth grade. VVho tell you
"Life sucks!" and everything they do is
"Boring!" VVho literally roll to the floor in
hysterical laughter when you separate the
prefix and suffix from the word "predic-
tion" and ask them for the root and what
it means. VVho wear short, skin-tight
skirts and leg-laced sandals, but carry
CONTENT AND LEARNING ISSUES
teddy bears in their arms. VVho use a
paper clip to tattoo km Morrison's picture
on their arm during quiet study, while
defending the merits of Tigger's personal-
ity in Winnie-the -Pooh. VVho send
obscene notes that would make a football
player blush, written in pink magic
marker, blasting each other for stealing or
not stealing a boyfriend, and sign the
note, "Love,
back." (Reiff, 1992, p. 90-91)
P.S. Please write
In their light-hearted intent, such
comments can spur teacher camarade-
rie, but they also can serve to remind us
of how easily this unique stage of
development might be misconstrued.
In fact, caution is in order since for a
variety of reasons beyond such lan-
guage, the intellectual character and
energy of young adolescents have often
been un(lerrate(l. Most certainly we
have nearly eliminate(1 reductive images
portraying young a(lolescents as "hor-
mones with feet," and yet the achieve-
ment question ought to call upon all of
us to make a more (leliberate effort to
acknowle (lge an (1 celebrate the incre (l-
ibly powerful intellectual character of
this age group.
As ~ travel around the nation, ~ have
ha(1 the (delightful opportunity to (lia-
logue with many young adolescents, and
I'(1 like to share with you that our young
people, from all ethnic anti cultural
affiliations, from all levels of income anti
from all levels of school competence,
repeate(lly (remonstrate that they are

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immersed in one of the most intellectu-
ally pivotal times in human develop
ment. ~ ask young people to share with
me the questions and concerns they
have about the world. Their musings
should remind us at how phenomenal
our educational opportunity is during
the middle school years. Let me share
a few of their recurring questions and
concerns:
· Will there ever be world peace?
· Can we clean up the environment?
.
Will men and women ever be equal?
· Will there ever be a black President?
.
What happens when you die?
· Why do people hate each other?
· Are millions of people in the world
really starving?
· Why is there so much hatred and
violence?
· Why does anyone have to be poor?
How can we cure AIDS and cancer?
· Is there life in outer space?
· Why are we here?
.
Young adolescents are philosophical,
investigative, renective, hypothetical,
and skeptical. They love to debate,
query, conjecture, moralize, judge, and
predict. They are filled with the joy of
self-discovery and the inevitable disillu-
sionment of world discovery. They are
paradoxical. While plagued with self-
doubt, they are armed with a heroic
invincibility (Elkind, 19841. In sum,
REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS
these young people are developmentally
ripe for intellectual growth.
But when they enter our middle
schools every day, not many find in their
classrooms a match for the intellectual
intensity their questions renect. For too
many, curriculum is not seen as excit-
ing, useful, meaningful, or helpful. In
numerous and lengthy focus group
interviews with students, ~ have found
their responses quite telling. ~ asked
one energetic sixth grade student what
he was learning and why. He responded
dutifully, "Latin America." ~ then asked,
' Why are you studying this? What's
really important about this topic?" He
was equally honest and said, "I have no
clue, but ~ think it's in the curriculum."
His peer offered, "I think we need to
learn it cause we might need it later on."
Another added, "No, ~ don't really think
so because my father is very smart and
successful, and ~ know he never uses
this stuff." Something huge is missing
in how students are experiencing
curriculum.
We shouldn't presume that it is only
those struggling students who raise
serious question with our curriculum. ~
asked a group of honor roll students in a
middle school to tell me about what they
had learned from the fall until January.
They couldn't recall much. In consider-
ing their plight of failed memory, one
eager student perked up with some
sense of enlightenment, "I think ~ know

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why we're slumped. Wewerein the
accelerated program and we went so fast,
we don't remember much." In the very
least, his comments affirmed my often
nagging notion that much of what we
define in schools as accelerated is merely
more content taught faster, more home-
work done hastily, and not much more
learned deeply. Clearly, the TIMSS
findings (Silver, 1998) illuminate this
"mile wide and inch deep" curriculum
problem that challenges our pedagogy.
That curriculum in all fields of knowI-
edge in middle school must be more
meaningful, challenging, anti engaging
is unquestionable. When observing
students and teachers at work, ~ dis-
cover that many of our students are
often not engaged in challenging learn-
ing experiences. ~ watched extremely
bored students sit in a well managed
classroom listening to a litany from
peers who were to each report to the
class on the content of an individually
rea(1 current event. This was an entire
class period of high disengagement and
notable disenchantment. This is where
an urgency for reform should be felt.
This still happens far too often.
THE REFORM PICTURE
While the middle school movement's
thirty year history has secured a place
on the reform map and has contributed
CONTENT AND LEARNING ISSUES
greatly to the overall improvement of
many middle level schools, the
movement's reform recommendations
and efforts have, ~ believe been more
successful in altering the climate and
structure of mi(l(lle level schools than
the curriculum and instruction our
young people have experienced (FeIner,
etal.,19971. Organizing smaller,more
personalize(1 learning communities,
commonly called teams, creating
teacher scaffol(ling anti support for all
students, emphasizing interdisciplinary
planning anti teaching, anti creating
more flexible sche(lules liberate(1 from
tracking, have without question, raised
teacher efficacy, encouraged profes-
sional (lialogue, re(luce(1 school ano-
nymity, improve(1 school climate, anti
even in some pockets, raised school
achievement. They have not always
resulte(1 in the (1ramatic shift in teaching
and learning that was ~ believe a bold
hope of the mi(l(lle school movement's
many advocates anti champions.
Perhaps the simplest explanation that
draws nods from many is that the
mi(l(lle school movement has (levote(1
too much of its energy and attention to
reforming the organizational character
of our mi(l(lle level schools. James
Beane (199Sa) woul(1 suggest that such
a situation with the state of reform in
mi(l(lle school teaching anti learning
was inevitable since we never fully
achieve(1 consensus on the goals anti

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purposes of the middle school curricu-
lum. Likewise, some speculate that
three decades of work devoted to the
creation of more humane schools has
resulted in soft attention to the intellec-
tual development of our young people.
Others argue that achievement, particu-
larly in mathematics, has been short-
changed by our advocacy and imple-
mentation of thematic teaching which
often highlights social studies and
language arts or (lisparagingly, reduces
mathematics to labeling correct mea-
surements on an interdisciplinary
exhibit. Still others ~ might say, quite
legitimately, have argued that mis-
guided interpretations of progressive
instructional methods have yielded
sloppy attention to intellectual develop-
ment and authentic and substantive
student learning.
Regardless of the complex puzzle of
causes that we are now facing, the
TIMSS results in mathematics are not
surprising. In the last three decades of
classroom practice, approaches to
mathematics instruction in middle
schools have not changed as consis-
tently and dramatically as some of us
might have hoped following the publica-
tion of the NCTM standards. The
islands of excellence are simply too few.
None of the recommendations for
middle school structural reform are void
of underlying theory about their rela-
tionshipw~thstudentlearning. While
REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS
not always clear to the public nor
consistently conveyed in professional
development, we should not be too
quick to blame the current achievement
conundrum in mathematics or any field
on the middle school concept.
Interdisciplinary teaming, for in
stance, remains at the core of middle
school reform, in large part because of
its research credibility to raise teacher
sense of efficacy a key element in
high performing classrooms. Team
organization has also been associated
with reduced school anonymity and
teacher collegiality, additional features
in safe and productive schools. More-
over, attempts at increased personaliza-
tion, in the form of Teacher Advisory
programs and similar middle school
initiatives, were fundamentally
grounded in the belief that the quality
of teacher-stu(lent relationships greatly
impacts student motivation anti perfor-
mance. That learning is a social en-
deavor, embedded in relationships is
not an assumption unique to middle
school philosophy (Glasser, 19921.
That curriculum was inten(le(1 to be
exploratory was not meant to suggest
that it could not also be (leman(ling.
That mi(l(lle schools were inten(le(1 to
be humane anti caring places was not
inten(le(1 to be antithetical to serious
mathematics education. In(lee(l, there
is a tremendous need to pursue and
refine these elements of the mi(l(lle

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school concept as we engineer a new
plan towards higher performing middle
schools with greater learning for all.
MIDDLE SCHOOL MATHEMATICS-
SEARCHING FOR SOLUTIONS
The standards movement is accompa-
nied by a wave of achievement panic
that threatens to diminish the focus on
learning in the middle grades. In this
panic context, the TIMMS data suggest
to some that classrooms have failed to
conduct sufficient skill and drill work.
Others cast the blame on thematic
teaching or detracking. Fortunately,
there are several themes that are
recurring in current conversations
among middle school advocates and
those interested in mathematics reform
which, when united, bring clarity and
perspective to some of the more emo-
tional attacks. Both groups call for a
curriculum that is challenging and
engaging for young people, offers
connections across disciplines, chal-
lenges students to apply knowledge,
putting mathematics to use, emphasizes
problem-centered learning, provides
opportunities for collaboration, and
seeks to en(1 inequitable practices like
tracking (Beane, 199Sb).
The earnest call to engage more
young adolescents in meaningful math-
ematics has led exemplary districts and
CONTENT AND LEARNING ISSUES
schools to pilot new programs and test
their own results. In Corpus Christi,
Texas, where standards-based reform is
ongoing, all but one middle school has
extended the invitation to take algebra
to all eligible 6, 7, ~ gra(lers. Eligibility
is still the sticky issue since aigebrafor
all (toes not mean all students are
guaranteed exposure to algebra con-
cepts by the Sth gra(le. In fact, while it
opens the door to early maturing,
frequently advantaged young people, it
still fails to embrace the very young
people we have missed all along.
In that same district one lone middle
school has employe(1 Connecte(1 Math-
ematics, a program developed out of
Michigan State University, anti the
student engagement anti learning
success they are observing are inspir-
ing. The manipulative and collaborative
nature of this curriculum approach finds
a place for varying levels of readiness in
a way ~ have not observed with tradi-
tional pre-algebra anti algebra ap-
proaches. Their story and the story of
other schools engage(1 in renective
practice and study will continue to offer
promise to our steady search for solu-
tions.
~ am frequently asked, "Should all
middle school students take algebra
before moving on to the high school?"
might begin by posing a clarifying
question, "As it is most often taught?" If
the response is "yes," then I am com

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pelted to answer, "no." We recognize
that many of our young adolescents are
not formal operational thinkers with a
strong logical-mathematical intelligence
and that algebra has traditionally been
taught to them as if they were. ~ am not
sure, however, that that is the right
question to ask; rather, shouldn't we be
asking, what kind of mathematics
should all young adolescents be learn-
ing to enhance their understanding of
the world and the role mathematics
plays in it?
Algebra as a course is problematic.
Middle school algebra for high school
credit is even more problematic. The
presence of algebra as a select course
with eligibility and teaching certification
requirements faithfully diminishes a
middle school's chances at academic
equity. When students are grouped for
mathematics instruction, they are
divided as well by race, economics, and
learning orientation. As middle schools
organize in small learning communities
to ensure the noted benefits of teaming,
students are grouped by mathematics
levels in ways that can result in tracking
and the reduction of mathematics
learning for non-aIgebra students.
These students deemed less ready or
able, travel apart from algebra students,
and may spend an entire year relearning
mathematics concepts many already
know, while they wait to enter "the
algebra course."
REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS
For young adolescents, meaning is
everything. In fact, human learning
involves meaning making, does it not?
Should we ted our young teens to endure
our current version of algebra because
they will increase their chances of going
to college or because then we can outper-
form our international partners?
If we are eager to embrace more
learners in mathematics education, these
suggestions will hardly inspire the
tentative. The same students with
involved parents or from advantaged
homes will be at our college doors while
those who wonder if there is life after
middle school or hope in life at all will
remain out of reach. Even among sup-
ported students we still must go further
as not one of my son's Sth gra(le hien(ls,
all in Sth grade algebra, can explain to me
why or how algebra is or even could be
useful in the world. Perhaps it is time for
a bold step to move towards creating in
middle schools, mathematics for life far
more challenging and meaningful than
what many currently experience in
algebra? Perhaps what all middle school
students should experience is the kind of
foundation algebra that few of us re-
ceived the kind that would make it
possible today for you to Lent the
many ways in which algebra is at work in
the world. Perhaps we might even be
able to recognize when we use it? Few of
us who (lo not teach mathematics can (lo
this well.

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What really stands in the way of true
reform in middle school mathematics
has less to do with education, however,
and more to do with politics. Math-
ematics is a subject area with social
status. To suggest that algebra not be
taught as a course reserved for "ca-
pable learners," is to invite a public
relations disaster of epic proportions
(Beane, 199Sa). In fact, in James
Beane's work with mathematics teach-
ers who have had success in teaching
mathematics in the context of curricu-
lum integration, many have begun to
schedule their mathematics as one of
the separate subjects in their pro-
grams, not because they think it is
sound educational practice, but be-
cause they would lose the rest of their
programs if they did not.
JOINING FORCES
We do have many important ques-
tions to consider: what is the purpose of
the middle school curriculum? Is it to
prepare students to be academic schol-
ars? To understand themselves and
their world? To make our students
score higher than students Tom other
countries anti with what assurance that
it translates into benefits in life and
work? To decide early who wait take
what path in schooling?
CONTENT AND LEARNING ISSUES
These questions also renect the major
points of deliberation such as whether
international test scores should shape
curriculum goals, whether all math-
ematics must be taught as a separate
subject, and to what extent higher
expectations ought to involve vertical
acceleration through mathematics areas
or application of knowledge to increas-
ingly sophisticated problems.
~ believe we have answers to what
constitutes best mathematics education.
Perhaps the really critical question we
nee(1 to a(l(lress is how can we make the
rhetoric of best practice a reality for
more of our young people. In answering
this question, we in fact push ourselves
towards a vision of mathematics e(luca-
tion that offers great hope for equity and
academic excellence.
REFERENCES
Beane, J. (1998a). The middle school under
siege. Paper presented at the National Middle
School Association's annual conference,
Denver, CO.
Beane, J. (1998b). Paper prepared for Middle
Grades Mathematics Convocation, September,
24-25,1998. Washington, DC.
Elkind, D. (1984~. All grown asp & no place to go.
Reading, MA. Addison-Wesley.
Felner, R. D. Jackson. A.W., Kasak, D., Mulhall,
P., Brand, S., Flowers, N. (1997~. The impact
of school reform for the middle years. Phi
Delta Kappan, 78~7), 528-532, 541-550.
George, P.S., & Shewey, K. (1994~. New evidence
for the middle school. Columbus, OH: National
Middle School Association.

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Glasser, W. (1992~. The quality school: Managing
sta~dentswitho?`t coercion. New York: Harper
Perennial.
Lipsitz, J. (1971~. Growing asp forgotten: A review
of research and programs concerning early
adolescence. New Brunswick, NI: Transaction
Books.
Lipsitz, J. (19843. Successful school foryoa`ng
adolescents. New Brunswick, NI: Transaction
Books.
REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS
Reiff, L. (1992~. Seeking diversity: Language arts
with adolescents. Portsmouth, NH:
Heinemann.
Silver, E.A (1998~. Improving mathematics in
middle school: Lessons from TIMSS and related
research. Report prepared for the U.S.
Department of Education, Office of Educa-
tional Research and Improvement.

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Kathleen Hart
Chair of Mathematics Education (retired, University of Nottingham, United Kingdom
In the United Kingdom compulsory
schooling starts at the age of five and
continues until 16 years of age. The
provision of free education continues for
another two years. The structure of the
school system varies and children can
proceed through infant school (age 5-7),
junior school (~-~) and secondary school
(~-16/~) or through a system in which
they change schools at 9 and 13 years of
age. So in England a child in the middle
gra(les is probably changing (or has just
changed) schools. In the first schools, the
teacher is a generalist and probably
teaches all the subjects the child meets
during a week. For the pupil, being
promoted to a secondary school (or even a
middle school) means that there are many
teachers to face in any one day. These
teachers tend to be interested in only one
curriculum subject. In the primary school,
it is likely that the teacher has tried to
present the curriculum through project
work, which might mean that the intention
was to exploit a topic (e.g., The Vikings)
for its possibilities to illustrate English,
geography, religion, art, science, and
mathematics. In the secondary school,
these subjects are allotted separate lime
slots, anti the teacher only teaches that
subject.
The institutional life in school
changes to a greater focus on formal
learning, and in mathematics the con-
centration is on competencies and skills
and their application. Often assump-
tions are ma(le concerning the reper-
toire of skills the child already pos-
sesses, and she may lose confidence
when it is shown her repertoire is
limite(l. A(l(1 to this the changes in
chil(lren's physical makeup anti the new
interests which occupy them, anti it is a
wonder that they learn anything.
FORMALISATION
MANIPULATIVE LINK
The influence of Piaget on e(lucational
theory has meant that much of the

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child's primary education has been
imbedded in the idea that prefix years
of age the child is operating at the
concrete level. Mathematics educators
for many years have interpreted this
stage as one which requires concrete
embodiments or manipulatives to
promote what are essentially abstract
mathematical ideas. The chasm be-
tween the manipulatives and the ab-
straction has not been addressed very
thoroughly. The research project
"Children's Mathematical Frameworks"
(CMF) sought to investigate the transi-
tion made by children when moving
from "concrete" experiences to a for-
mula or mathematical generalisation.
Teachers who were pursuing a masters'
degree in mathematics education
enrolled for a module which required
them to:
a. Identify a topic which they would
normally introduce with the use of
manipulatives, which experience was to
lead to a formula, algorithm or other
mathematical generalisation.
b. Prepare a series of lessons and
teach them to a target class.
c. Allow the CMF research team to
interview six children, before the
teaching started, just before the
formalisation took place, just after it,
and three months later.
d. Alert the researchers to when the
"formalisation" lesson or acceptance of
MIDDLE GRADES MATHEMATICS
the rule would take place and allow the
lesson (s) to be observed and tape
recorded.
e. Interview two other chil(lren in
the class and report on the responses.
Additionally, an analysis of the tape
recorded lesson would be written (the
transcript of the recording being sup-
plied by the researchers).
Topics which were included in the
study, fulfilling the description of con-
crete embodiments leading to
formalisation, were area of a rectangle,
volume of a cuboid, subtraction of two
and three digit numbers With decompo-
sition, the rule for fractions to be equiva-
lent, the circumference of a circle, and
enlargement of a figure.
The advice given to teachers in teach-
ing manuals etc., often describes the
experiences the children should have
and then implies (or even states) that
"the children will come to realise" the
formula. In practice, it seems that a few
children in a class might come to the
realization, and the rest be encouraged to
accept the findings of their fellows. The
teacher feels that time is short, and the
class must move on. Part of the three
month follow-up interviews was to ask
the pupils for the connection between the
two experiences, concrete and formal.
Only one of the interviewees (out of 150)
remembered that one experience led to
the other and provided a base for it.

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Most of the replies are summed up by
the girl who said "Sums is sums and
bricks is bricks." The forgetting would
be unimportant if the concrete experi-
ence (which is often arduous and time
consuming) had resulted in a successful
use of the formalisation, but it had not.
The observations of lessons and analysis
of the transcripts of what the teacher said
brought to light how very disparate were
the views of teacher and pupil. The
teacher knew the mathematics, knew the
formula or rule, and had devised a set of
"manipulative" moves to convince the
child of the truth of the rule. The child
did not know where the manipulations
were leading and to him/her a red brick
made from two centimetres of wood was
exactly that. The teacher might refer to
it as "x, 2, 4" and could even say "let us
pretend it is 17."
Figure 1.
Terence's Diagram for Equivalence
CONTENT AND LEARNING ISSUES
JOINING THE GROWN-UPS
From the observations in CMF (an(1
some subsequent research), it was plain
that teachers and children embarked on
a voyage of (liscovery to a place well
known to the teacher. None of the
teachers observed (some 20 experi-
ence(1 practitioners) explaine(1 why the
pupils woul(1 want to abandon bricks,
naive methods, anti even invented chil(l-
methods in favour of the formalisation.
Nobody explained the power of the new
knowledge. The nearest statement to a
reason for its adoption was "you do not
want to carry around bricks for the rest
of your life." The teacher's attitude was
one of frien(lly guidance, more in the
sense that the lessons were a review of
something we already knew rather than
an introduction to the complete un-
known. Consi(ler how few teachers
(lraw on the boar(1 accurate subdivisions
when partitioning a circular (lisc into
equal fractional segments. The illustra-
tion is produced free-hand and quickly
split into sections. Little wonder that
Terence pro(luce(1 this set of (1iagrams
when he was trying to convince the
interviewer that 9/27 = 3/9 (Figure 11.
Andrew was in a group learning
subtraction when the teacher pro(luce
a three (ligit subtraction which resulte
in zero in the hundreds place. The
ensuing conversation with the class of
eight pupils was as follows:

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(Note: T = Teacher; P = Pupil):
P: And that wouIc3 be one huncirec3
take away one huncirec3 is nothing.
T: Is nothing, so c30 I put that there?
P: No.
T: Shall I put that there? Who thinks I
shouic3 put that there? Who doesn't think
I shouIc3 put that there? Well, I mean, you
can, but if I was to ask you to write clown
99 . . . in your books, just write clown 99,
you wouIcin't write clown 099 wouIc3 you?
P: No.
T: You wouic3 just write the 99,
wouicin't you? So we don't really need to
put that there. 1 take away 1 leaves you
with an empty space, so we might as wed
leave an empty space, okay?
Andrew's attempt at 304-178 gave the
answer 2 and his argument went as
follows:
A: Say if you said, four take away 8,
it's 4. You've got 4 and you can't take 8
from 4, so there's nothing there and . . .
nothing take 7, you can't c30 that and 3
take away 1 gives you 2.
T: I see. So take 4 away 8 I can't c30,
right. So c30 I write anything underneath
there or c30 I not bother?
A: Not bother.
MIDDLE GRADES MATHEMATICS
He had absorbed the "don't bother"
but not when to use it. There were other
instances in the interviews of children
selecting a specific part of a teacher's
statement and generalising incorrectly or
of remembering the one erroneous
statement the teacher had made.
FRACTIONS AND DECIMALS
The middle grades are the years
when the elements of arithmetic cease
to be exclusively whole numbers, and
much energy and time is spent on the
study of fractions and decimals. There
has been a lot of research on children's
understanding of these "new" numbers.
Generally most eight-nine year olds can
recognise and name a region as 1/2, 1/4,
1/3, 1/~; fewer recognise that a region
split into twelfths can also be labeled in
sixths. Far fewer chil(lren can success-
fully carry out operations on fractions.
The mode} for introduction currently in
most of our textbooks is that of regions
(square, circle, line). This enables us to
talk of shares, but the result is a tangible
amount (slice of pizza, cube of choco-
late) which (toes not neatly fit within the
operations of a(l(lition, subtraction,
multiplication anti (livision. How can
you multiply two pieces of pizza? The
other meanings of a/b are often not
a(l(lresse(1 separately in school text-
books, anti the chil(1 is expecte(1 to infer

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division or ratio from the same region
model. The result is confusion and a
heavy reliance on rote-learned rules.
Many children reject the whole idea of
noncounting numbers and attempt to
deal with secondary school mathemat-
ics without them. The research project
"Concepts in Secondary Mathematics
and Science" (CSMS) obtained data by
both interviews and tests. A representa-
tive sample of 11-16 year olds was asked
to give the answers to various division
questions, and they were told that if the
question was "impossible" they should
say so. Table ~ shows some results.
A parallel test had required pupils to
write a story to illustrate ~ - 4 and
128 - 8. Very nearly all the responses
involved the sharing of sweets among
hiends. This interpretation of the divi-
sion sign is in connict with 16 - 20 as
there are obviously not enough sweets
for the Fiends to share. Do teachers
redefine the operations to accommodate
fractions and decimals? The middle
grades is when children are trying to graft
new concepts on hopelessly inadequate
foundations put in place for counting
numbers. Algebra is likely to be intro-
duced during these years and viewing it
as generalised arithmetic seems fraught
with difficulties. In algebra, we need to
sped out all connections among Me
numbers now represented by letters, and
in ari~medc Me aim is to carry out Me
operations anti to obtain a result as
quickly as possible. "x +y" stays as such
and cannot be processed to become xy
whereas we find it unwieldy to work wad
5 + 3 and replace by ~ as soon as possible.
Collis (1975) (lescnbe(1 a level of algebraic
un(lerstan(ling as "Allowing Lack of
Closure" (ALC). When a chil(1 can accept
anti even work wig (x +y), a significant
step has been taken.
Table 1. CSMS Results to Division Questions
Large Survey Results
Divicleby20 1.21.0 1.A 1 rem A Impossible
(i) 2A (n=170) 11-12yr 9%7% 8% 12% 15%
(n=2~0) 1~-15yr 3~%1% 15% 3% 6%
0.80.0 0.16 0 rem 16 Impossible
(ii) 16 11-12yr 7%2% A% - 51%
1 A-1 5 yr 36%- 6% - 23%
CONTENT AND LEARNING ISSUES

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MATCHING THE MATHEMATICS
TO THE CHILD
It has long been known that from any
class lesson, the child participants take
away very different pieces of knowI-
e(lge. In CMF a group of eight pupils,
all thought by the teacher to be "ready"
for subtraction and all taught in the
same way, were found to have very
different paths to success. A child's
success depends on what was known
before Now many of the pre-requisite
skills are in placed; how much of the
current content matter is understood;
his attention span (was he even in
school, absences matter) and the
confidence with which the mathematics
is approached (does the child fee} in
control of the mathematics content or is
it magical. A child cannot be confident
if all the mathematics exercises he does
are marked incorrect. By the middle
gra(les it is likely that any group con-
tains a number of "low attainers" who,
without positive action, are unlikely to
become even "average attainers."
The curriculum development project
"Nuffield Secondary Mathematics" was
designed to provide suitable material for
all attainment levels in the secondary
school (ages 11-16 years). The books
were: a) Topic books; short "content"
orientated material in four sets-
Number, Space, Probability and Statis-
tics, and Measurement; b) Core books;
MIDDLE GRADES MATHEMATICS
books of problems for an entire year to
allow groups of mixed attainment to
work together applying their mathemat-
ics; anti c) Teachers gui(les; very full
information for teachers. To find where
to start the Number strand, children in
primary school (some 100 pupils),
including some who were identified as
(lisplaying "special needs" were teste
with items that researchers had previ-
ously used with six year olds. Follow-up
interviews disclosed that there were
pupils about to enter secondary school,
who had very limited number skills. A
list of pre-requisite skills was drawn up,
and we stated that to start on the Num
her books, children had to demonstrate
that they could do the following:
· Arrange car(ls showing configura-
tions of (lots for ~ to 6 in order.
· Give the number before anti after a
written two-digit number.
· Count on (rather than count from 1)
when given two strips of stamps.
· Write correctly two-(ligit numbers
when they are rea(1 out (oral).
· Put written numbers, less than 25, in
order (written).
· Interpret the wor(ls "more" anti "less"
when given two sets of (lots
· Count a pile of coins Hess than one
pound) accurately, taking account of
the (lifferent face values.
· Choose the correct single coin for a
purchase of 45p.

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Practising secondary school teachers
during in-service courses were shown
the contents of Level One Number and
asked if they had pupils who could only
"do this." Usually they assured us that
none of their pupils knew so little, but
later we were bombarded with requests
for the material at this level. In the trial
schools there were usually about ten
11-12 year olds who needed it. We only
found one pupil in a normal school for
whom the work was too difficult. The
material was put through several, wed
regulated trials and rewritten if it proved
too difficult for the pupils using it. The
intention was that the child should
experience success and so become
confident in mathematics, no matter how
limited. An early result was that al-
though the first book took perhaps three
months to complete, the second took
much less time. As confidence built, so
did the speed with which the child
worked. The classes using this early
material were usually small, and the
pupils worked in pairs or groups of three
with a lot of teacher help. No child
progressed to the next book until he/she
had demonstrated that the content of the
previous book had been absorbed by
passing a test at the 80% (or better) level.
No test was given until the teacher was
sure that the child would pass because all sufficient.
the book had been understood. Failing a
test helps neither teacher nor chil(l. This
concept was very difficult for some
CONTENT AND LEARNING ISSUES
teachers because they assumed there
would always be failures. We were
closely involved in all the trials and often
marked the tests. In one school, the
teacher agreed that the reason six
children hall not reache(1 "mastery" level
in the test was because they had not
covered the entire book but only part of
it. This must perpetuate a situation
which is bound to be deficient half
learned mathematics grafted onto holes
in knowledge. The teacher explaine(1
that she could not wait for these pupils.
CONSTRAINTS AND BELIEFS
Running parallel to any new curricu-
lum effort, and having an unseen but
powerful influence on it, are the con-
straints and beliefs of the general
populace, politicians, headmasters,
publishers, and even classroom teach-
ers. Some of these are listed here:
1. There are certain topics which
every chil(1 should be taught.
2. There are specific topics that
every chil(1 should have learne(1 before
a certain time in his/her life.
3. A certain amount of time spent in
school on mathematics lessons is
4. A certain amount of material ([books,
worksheets or scheme) is enough.
5. Mathematics is (difficult.

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We should be wary of these beliefs
because they are very strongly held
and usually backed up by appeals to
"raising standards." In the UK during
the summer of 199S, the Minister of
Education boasted that the number of
pupils passing the school-leavers'
examination in mathematics had
fallen, in order to claim that standards
had been maintained. The expecta-
tions the community has of pupils
vary from country to country and
what may seem obvious to a Japanese
writer, is not obvious to an Italian
observer. Howson (1991) published a
list of ages at which specific math-
ematics content was presented to
pupils.
An excerpt is shown in Table 2. There
is obviously no "obvious" age for the
introduction of a topic.
CONCLUSION
Mathematics in the middle grades is
still "Mathematics for All," although the
move is towards a formalisation of the
subject. Failure to understand destroys
the child's confidence so any introduc-
tion of new concepts, such as numbers
which cannot be used for counting
objects, must be built up carefully and
with few assumptions on the part of the
teacher. Learning mathematics is a
series of leaps, so it is good to know that
the ground from which you take off is
soli(l.
Table 2. Age of Introduction of Content (aclaptecl from Howson, ~ 991~
Belgium France Italy Japan England
Decimals9 9- 11 8- 11 8 9
Negative numbers8 11-12 11-1A 12 9
Operations on these1 2 1 2-1 3 1 1 -1 ~1 2 1 3
Fractions7 9- 11 8- 11 8 11
Use oflelters12-13 11-12 11-1A 10 13
REFERENCES
Cockcroft Committee of Inquiry Into The
Teaching of Mathematics in Schools. (1982~.
Mathematics counts. London: HMSO.
Collis, K (1975~. Cognitive development and
mathematics learning Chelsea College,
P.M.E.W.
MIDDLE GRADES MATHEMATICS
Hart, K (Eddy. (1981~. Children's understanding
Mathematics: 11-16. London: John
Murray.
Howson, G. (1991~. National cavrric?`la in
mathematics. Leicester, UK: The Mathematical
Association.
Johnson, D.C. (Ed). (1989~. Children's math-
ematicalframeworks 8-13: A study of classroom
teaching Windsor, UK: NFERNelson.

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~]
Working through a problem in the role
of student can serve as a springboard for
a discussion of the issues around content
and learning mathematics. Contrasting
student solutions with adult solutions
(Appendix 4J furthergrounds the conver-
sation in a situation reading student
responses that is in fact, part of the
practice of teaching Marcy's Dots
~ ~_~
(Figure ~J from the 1992 National
Assessment of Educational Progress grade
test provoked a variety of responses
rangingirom concern over the clarity of
directions to surprise at the many and
diverse ways students found their solu-
tions. Common themes that emerged
from the group discussions are described
below.
Figure ~ . Marcy's Dots Problem from the ~ 992 National Assessment of Eclucational Progress
A pattern of clots is shown below. At each step, more clots are aclclecl to the pattern. The number of clots aclclecl at
each step is more than the number aclclecl in the previous step. The pattern continues infinitely.
( 1 St step)
(2n~ step)
(try step)
·
·
2 Dots
6 Dots
1 2 Dots
Marcy has to determine the number of clots in the 20th step, but she does not want to clraw all 20 pictures and then count them.
Explain or show how she could clo this and give the answer that Marcy should get for the number of clots.
Note: See Appendix 4 for sample student solutions and the guiding questions for the discussion.

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STUDENT WORK
"Student work helps teachers think
about how students learn mathematics
and about the depth of their own under-
standing." (Participant comment)
Studying how students come to learn
mathematics by using a "site of prac-
tice," an activity that is something
teachers do as part of teaching, led
some discussants to conclude that the
richness of student thinking about a
problem accompanied by student
solutions can be buried in the raw
statistics reporting student achieve-
ment. The variety of strategies used by
the students in their solutions charts,
recursive rules, formulas, listing all of
the cases, drawing diagrams paralleled
the strategies used by the adults.
Discussing how students used their
understanding of the mathematics to
select an approach led to a discussion
about the reasoning used by the adults,
and in fact, led to some stimulating
mathematical discussions. In some
cases, there was concern over the lack
of consistency between the work stu-
dents did and their description of what
they did. This was attributed to a lack of
communication skills on the part of the
students, although learning to commu-
nicate is an important middle grades
topic.
An effective strategy to promote
student learning could be to have
SUMMARY OF SMALL GROUP DISCUSSION
students themselves learn by using
other student work. The question of
quality answers vs. quality thinking
became an issue, however. How do
teachers reward and reinforce correct
thinking even though the desired
solution is not presented? Groups
identified non-mathematical causes for
student errors, answering the wrong
question, not reading carefully, or
jumping to conclusions, as well as
mathematical reasons such as identify-
ing the wrong pattern. A si(le effect of
the analysis of the (liversity in the
student work was a reminder to the
teachers not to impose their solution
method on their students.
THE MATHEMATICS
The question of teacher knowledge
and capacity to deal with problems such
as Marcy's Dots is a serious one. There
was a strong feeling that many middle
grade teachers do not have the neces-
sary background to deal with some of
the broad mathematical and algebraic
concepts involved in the problem:
variables and an introduction to sym-
bols, functional relationships, linearity,
sequences and series, recursion. The
problem links algebra and geometry, is
multi-step involving logical thinking, and
leads to making generalizations. The
perception that teachers are not pre

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pared to teach this kind of mathematics
was reinforced by studying the algebra
portion of the draft version of the
revised National Council of Teachers of
Mathematics standards: Principles and
Standards for School Mathematics:
Discussion Draft. The issue of certifica-
tion for teaching mathematics in the
middle grades, the nature of preservice
programs, and professional develop-
ment were repeatedly identified as
critical in helping teachers move beyond
their comfort with number to other
mathematical strands that should be
part of the middle grades curriculum.
There was also strong agreement that
content knowledge is not enough;
teachers must learn how to help stu-
dents bridge from the concrete to the
abstract.
ALGEBRA IN THE
MIDDLE GRADES
The issue of whether ad eighth grad-
ers are (levelopmentally really for aIge-
bra and how to position algebra in the
learning environment of the child raised
more questions. There was a consistent
belief that the study of patterns was
important, but there was tension over
how to move from the specific to a
CONTENT AND LEARNING ISSUES
generalization with ah students. The
nature of the problem allowed students
with different abilities and understanding
to find a solution, an important feature
for good problems. It is important for
students to see each others work and
then come to consensus on an effective
way to solve the problem at the most
abstract level that the students are
developmentally capable of understand-
ing. Common beliefs were that the use
of a problem without a context lacks
motivation for students and that prob-
lems should be relevant and real to
engage students. Students should be
able to see where a problem is going and
how it connects to other areas they are
studying. They need to understand why
formulas and generalizations are impor-
tant, as well as how to think about and
use them appropriately.
ORGANIZATIONAL ISSUES
Two organizational issues surfaced as
barriers against practicing teachers
approaching any such problem in a
thoughtful and analytic way: the lack of
time during their school life to engage in
this kind of thinking and the current
emphasis on testing, where most of their
energy is concentrated on what is being
tested or the need for accountability.