As
Piaget stated, "Children have real understanding only of that which they
invent themselves, and each time that we try to teach them something too quickly,
we keep them from reinventing it themselves." (Papert,
2001, p. 2) And customarily, we teach math too quickly to our students in
the
To
build a tall building, one requires a strong foundation; and, to get the
American students at a height that the reformists hope them to achieve,
students need to start constructing the foundations of understanding at the
earliest. This means that we have to start at the base with elementary students
and elementary teachers if we want to meet the raised expectations for middle
and high school mathematics. Also, it is a fact that “Age increases at a
constant rate but cognitive level …tends to follow an uneven fast, then slow,
trend.” (Stokes, 1990, p. 313) Furthermore, we cannot
predict the periods of cognitive growth. As such, if we don’t provide our young
students with the right kind of intervention and experience in their early
years, we might deprive them of some very vital cognitive growth opportunities.
As Conference Board of the Mathematical Sciences (CBMS) recommendations (2001) summarize:
It
is during their elementary years that young children begin to lay down those
habits of reasoning upon which later achievement in mathematics will crucially
depend. Thus, for example, it is unrealistic to expect students who failed to
develop early an understanding of how to manipulate arithmetic expressions to
later manipulate algebraic expressions with confidence. And those students who
have never had experience with decomposing and recomposing shapes in their
early education are unlikely to attach meaning to the succession of assertions
in typical proofs in Euclidean geometry…The power to reason mathematically is a
natural human capacity. Young children enter school already curious about
number and size, and with ideas about how to join, remove, and split
quantities. Mathematics instruction in the elementary years can---should---be
designed to cultivate this curiosity. Encouraged to solve problems, children
become aware of their ideas; and as they learn to analyze their own, their
classmates’, and their teacher’s thinking, these ideas become more refined and
many-sided. It is during these early years that young students lay down those
habits of reasoning upon which later achievement in mathematics will crucially
depend.
Hence,
it is our duty as educators to make sure that we present young minds with every
opportunity to make exciting discoveries and grow intellectually into students
who have the understanding to appreciate the value of mathematics and enjoy the
challenges that come with it. To make an effect, we need to introduce reforms
at the beginning because anything later than the earliest might be too late.
Teaching
elementary mathematics is extremely challenging and prospective elementary
school teachers must be prepared to teach everything from counting to algebraic
thinking (Franke, 2000). They must not only understand the
mathematics they are to teach but also be able to engage students in that content.
The subject knowledge is indispensable because only teachers who have a deep
understanding of content are capable of “making connections, promoting
discovery, evaluating alternative responses, and providing experiences that
promote higher-level thinking skills” (Cain, 2000). On the
other hand, content knowledge is not enough because we expect teachers to do
more than walking their students step-by-step through an algorithm. As Franke (2000) points out:
Knowledge-in-action
demands that teachers know how to use their knowledge of the content to enhance
the understanding of their students. Teachers need to know what question to ask
when a student tells them that 7 =3 +4 is not true. They need to know which
problem to pose next and what a student could say to demonstrate understanding.
That
is, we need elementary teachers to develop an attitude towards learning and
teaching mathematics that will help them communicate the truth that mathematics
is not “a succession of disparate facts, definitions, and computational
procedures to be memorized piecemeal” (CBMS Recommendations,
2001). Rather, we need to provide elementary teachers with a program that
will enable them to become teachers with “detailed knowledge of students’
thinking, a way of organizing that knowledge, and a view that this knowledge is
theirs to add to, challenge, and adapt” (Franke, 2000).
Let’s look more closely into the three sets of expectations as outlined by the
CBMS, NCTM and Achieve (Foundations for Success).
The
CBMS recommendations (2001) sum up the qualifications of an elementary teacher
as follows:
Teaching
elementary mathematics requires both considerable mathematical knowledge and a
wide range of pedagogical skills. For example, teachers must have the patience
to listen for, as well as the ability to hear, the sense---the logic---in
children's mathematical ideas. They need to see the topics they teach as embedded
in rich networks of interrelated concepts, know where, within those networks,
to situate the tasks they set their students and the ideas these tasks elicit.
In preparing a lesson, they must be able to appraise and select appropriate
activities, and choose representations that will bring into focus the
mathematics on the agenda. Then, in the flow of the lesson, they must instantly
decide which among the alternative courses of action open to them will best
sustain productive discussion.
Plainly
speaking, the CBMS recommends that the elementary teachers should be highly
aware of the numerous connections in the mathematical ideas. They should be
flexible in drawing upon their personal knowledge of the subject and help
redirect students as they wander off many tangents to focus back on a
productive discussion about the math involved. Therefore, the teachers should
have ample patience to listen and understand the reasoning of their young
scientists so that they can build on a student’s conjectures rather than giving
them recipes of the right way to solve a problem. In short, the teachers should
act like a catalyst in a class full of experimentation and exciting
discoveries.
In
addition, there are some expectations on behalf of the National Council of
Teachers of Mathematics (NCTM). Although the document doesn’t explicitly
outline the requirements for elementary teachers’ certification, it does
present the following set of expectations for teachers in general and some
aspirations for the elementary teachers in specific (NCTM,
2001):
In
the elementary grades, convincing students that they can do mathematics and
helping them enjoy it are important goals.
Elementary school students need at least an hour of mathematics instruction
each day. The decisions teachers make in the classroom about how to offer all
students experiences with important mathematics and how to accommodate the
wide-ranging interests, talents, and experiences of students are essential
to giving all students access to mathematics. Although many matters bearing on
their classrooms are beyond teachers' sole control, they need to take the
initiative in discussing trends and opportunities in mathematics education with
administrators. Mathematics teachers can foster reinforcement of their efforts
by families and other community members by maintaining dialogue aimed at the
improvement of mathematics education. To do all of this well, teachers need to
understand their mathematical goals and their perspectives on mathematics
education and be able to articulate them in compelling ways. They should
constantly evaluate curricular materials and offer suggestions to
teacher-leaders and administrators, and they should find ways to be involved in
choosing the instructional materials for their school or district.
That
is, the NCTM guidelines encourage teachers to participate as active members of
the math community outside their classrooms and lay higher standards for
involvement of all teachers, including the elementary school teachers.
Furthermore, it encourages elementary teachers to challenge each and every
student according to his or her personal intellectual capabilities. To meet
this demanding expectation, as children grow at a variety of rates in terms of
cognitive development, teachers need incredibly high energy and patience levels
to monitor and challenge each individual. Also, the NCTM suggests a better
communication amongst the parents and teachers to create a support system for
the growing minds at home as well as school. This again calls for teachers to
go farther than the class work and reach the primary educators of any child,
that is, his or her parents. Finally, the council stresses that teachers need
to make it a priority to encourage the natural curiosity of young students and
guide them to a path of discovery where they can unravel the coexisting beauty
and power of mathematics at their own pace.
Yet
another set of expectations that are parallel to the NCTM standards, are laid
out by the consultation draft of Achieve, Inc. titled Foundations for Success
(FFS). The Foundations for Success is mainly written for the middle school
curriculum but it includes a few references as to what the elementary students
should be prepared for in order to successfully meet the Achieve aspirations
for math in the middle school. The specific expectations that refer to
elementary students and not elementary teachers per se are as follows (FFS, 2001):
Students
need to enter the middle grades with confidence that mathematics is a source of
useful tools for solving interesting problems. To build confidence and
enthusiasm, students need strong preparation from kindergarten to grade five,
including:
·
Fluency
with manual computation and mental estimation;
·
Experience
visualizing and drawing geometric objects;
·
Practice
formulating mathematical questions from various contexts;
·
And,
plenty of opportunities to explain and critique mathematical thinking and use
mathematics to solve problems.
Elementary
school students need to become fluent with the basic computations of
arithmetic, and they also need to understand why these procedures are valid and
what concepts they represent. Thorough understanding grows best from extensive
hands-on experience-in measuring and counting, exploring common geometric
objects and in representing data in different forms. It is not enough to focus
just on computational and procedural skills because students’ ability to reason
mathematically depends on a deep understanding of central mathematical
concepts. In turn, procedural skills provide firm support for conceptual
understanding.
The
Achieve expectations are in line with the thought that elementary students need
to develop a thorough mathematical understanding of the elementary concepts;
however, they do not specifically dictate what the teachers need to do so that
they can assist their students in reaching that understanding. Instead, they
provide numerous examples and sample solutions that primarily focus on
“concepts that need clarification or are difficult to teach” (FFS,
2001). The illustrative problems also attempt to clarify subtleties in terminology
through specific examples.
As
a whole, the expectations for elementary teachers are high and demanding. To
enable and support the elementary teachers in completing the crucial task of
founding the floor of mathematical understanding, every community needs to
provide their elementary teachers with not only the right education and enough
preparation but also with generous incentives and genuine appreciation.
In
the words of Leitzel, “The mathematical preparation of elementary school
teachers is perhaps the weakest link in our nation’s entire system of
mathematics education.”(Hungerford, 1994, p. 15) (Leitzel,
1991). In most states, teachers in grades k-6 are not mathematics specialists.
In fact, only 7 percent of elementary school teachers majored or minored in
mathematics or mathematics education. Furthermore, 40 percent of the elementary
school teachers report that they do not feel qualified to teach the content
that they teach (Stiff, 2000). Along the same lines, in a
report submitted by the Committee on the Undergraduate Program in Mathematics
(CUPM) in 1963, it was found that of the colleges that reported, 55.6 percent
offer no mathematics courses specifically designed for prospective elementary
school teachers (Hardgrove, 1963, p. 872). The scene has not
improved significantly since the 1963 report by CUPM. In 1998,
There
are multifaceted issues contributing to the above-sketched gloomy state of
affairs. One simple reason is that unfortunately, neither the students
(prospective elementary teachers) nor their instructors have any immediate
reasons to change. Moreover, the prospective elementary teachers generally have
“weak mathematical backgrounds and a high level of mathematical anxiety” as
students (Hungerford, 1994, p. 15). Many a times, the
elementary teachers have had a bad learning experience with mathematics and
claim that they hate math, they couldn’t learn it and they can’t teach it (Cornell, 1999, p. 225). And, as a whole, the entire education
system is currently playing a circular blame game. The college professors tend
to blame the high school teachers and they, in turn, blame the elementary
school teachers for the poor mathematical preparation of incoming students.
However, elementary teachers are trained by the same college professors who
complaint about the incoming students (Hungerford, 1994, p. 15).
What’s more, many instructors who teach courses for elementary teachers do so
unwillingly because of the unattractive structure of courses that summarize
subject matter for grades k-8 in approximately six semester hours (p.
15). Furthermore, there are very few incentives for instructors to invest
their time in improving teaching methods; they are much highly rewarded for a
job well done at research instead.
Another
issue that cannot be ignored is the fact the instructors for whom the subject
comes easily, lack respect for their math-anxious and math-avoidant students (Andrews, 2000). The college teachers often riddle their
lectures with disrespectful remarks such as “It’s easy”, “Couldn’t be simpler”
and so on. Also, the teachers’ assumptions of student’s knowledge can make it
difficult for the teachers to recognize that “seemingly simple,
self-explanatory processes may be complicated to others” (Cornell,
1999, p. 226). And, further use of
obscure vocabulary, incomplete instruction by skipping sub-steps for
mathematical procedures, overemphasis on rote memory and presenting math in
isolation of the real world, can lead the math-anxious students to become
frustrated and give up trying altogether. Finally, it is not a surprise that
“Elementary teachers who don’t know much mathematics, who have little interest
in what it means to do mathematics, and who are afraid of mathematics, are not
likely to engender positive attitudes toward mathematics in their students. Yet
these are the kind of teachers that current system is geared to produce.” (Hungerford, 1994, p. 16)
Undoubtedly,
there is a lot that needs to be done. However, before carrying out any
suggestions, it is foremost important to note that for successful
implementation of any plans, it is crucial that the people involved in
elementary teachers’ preparation develop a right attitude towards their
education. There is a critical need to increase the awareness and respect in
the math society for the kind of educational training that the elementary
teachers need. As Angela Andrews (2001) claims in her paper,
it is always easier to blame the victims than to teach them; and, the majority
of teacher educators are accustomed to blaming the weaker math students rather
than helping them break free of the fears developed towards learning
mathematics in their earlier school experiences. This negative attitude of the
instructors disables them from appreciating the potential “gift of the
math-anxious teachers” that are already motivated by their own struggles and
want to ensure that none of their students have the same difficulties. Andrews’
remarks are supported by the CBMS recommendations as well. The CBMS
(2001) recommendations state that:
Too
many students preparing for elementary teaching have been less than successful
mathematics students, and even those with good grades often doubt their
competence. Understandably, readers of this document may feel dismay at the
prospect of working with such math-anxious, if not math-phobic, undergraduates.
However, those who work with them can testify that, once these prospective
teachers experience their own capacities for mathematical thought, their
anxiety is transformed into energy for learning. In taking responsibility for
the kind of instruction for elementary teachers envisaged here, mathematicians
are invited, in effect, to re-enter the world of the naďve mathematical
thinker. The recognition that the "unsophisticated" questions teachers
pose do raise fundamental issues should inspire instructors to find contexts in
which these can be addressed fruitfully. This means, at least initially,
approaching the mathematics from a concrete and experientially based, rather
than an abstract/deductive, direction. Isn't this the way each of us starts our
individual journey into the world of mathematics?
Evidently,
it is vital for the math educators to “re-enter the world of naďve mathematical
thinker” in order to prepare the teachers who will work with young minds at the
beginning of their journey to construct and develop mathematical understanding.
Finally, it is important to note that attitude adjustment
is required on the behalf of education department as well, because effective
change won’t be possible unless the education college requires its elementary
majors to participate in the reformed program. Many education faculties do not
value mathematics to the same degree as mathematicians and are satisfied with
the present situation; therefore, in this case, it is essential for the math
department to “sell” mathematics to others (Hungerford, 1994, p.
19).
Secondly,
there is a need for efforts at community and state levels to develop a support
system to help the in-service teachers further their education and the
pre-service teachers get the right training to meet their job’s high and
increasing expectations. To list a few steps needed, “district and school
officials must foster working conditions that allow teachers’ knowledge to grow.
Curriculum and assessment developers must create appropriate instructional
materials-moreover, publishers understand that developing such material takes
time and must be willing to publish them. Finally, when considering curricula,
members of state and local boards must know routes to mathematical
understanding, as well as the nature of this understanding, when selecting
tests for students and setting criteria for teacher certification.” (Kessel & Ma, 2001) Also, we need a sufficient number of
math majors to enable the departments to offer a variety of interesting courses
and to obtain adequate support form the administration. Mathematics departments
need to devote commensurate resources to designing and offering courses for
teachers. They need to value and properly reward faculty members who are
heavily involved in teacher education. Mathematics departments engaged in
teacher preparation should establish a committee on teacher education that can
serve as a vehicle to engage more faculty in teacher preparation (Tucker,
2000).
Additionally,
we need to increase the math requirements for elementary teachers’
certification. The CBMS specifically recommends at least nine credit hours in
mathematics at college level to be a part of the elementary certification. The
three courses may choose to cover topics on number and operations, on geometry,
and on algebra, and respectively, selecting activities related to data to
support content in each of these areas. Or, some institutions may choose other
themes to define their courses that may integrate content from number, algebra,
geometry, and data into each of the three courses. In either case, content and
methods courses should be coordinated. However, it should be realized that
“simply increasing the number of required credit hours is no solution - courses
that allow students to get by using the same stratagems that got them through
K-12 just perpetuate the problem” (CBMS, 2001). Also, in
addition to more courses from those already offered for other audiences, some
new courses specifically designed for this audience might be needed. Moreover,
it won’t be enough simply to design new courses or select new texts.
Instructors in these courses must be encouraged to adopt teaching styles other
than the passive model: “I tell you what mathematics you need” and replace it
by an active one: “With my assistance, you explore, discover, or construct the
mathematics you need” (Hungerford, 1994, p.18).
To
repeat, the challenge is to work from what teachers do know---the
mathematical ideas they hold, the skills they possess, and the contexts in
which these are understood---so they can move from where they are to
where they need to go. For their instructors, as we have seen, this means
learning to understand how their students think. The habits of abstraction---of
compression---and deductive demonstration, characteristic of the way
mathematicians present their work, have little to do with the ways children
build their mathematical world, experientially, modeling concepts on actions---counting
out, dividing up, comparing heights or ages. ... Mathematics courses for
teachers must aim, first of all, at helping them develop ways of giving meaning
to the mathematical objects under study, only then moving on to higher orders
of generality and rigor (CBMS Recommendations, 2001).
Furthermore,
we have to recognize the limits of time and face the fact that elementary educators
are generalists, not specialists. Therefore, it is more important to help them
understand and teach fundamental mathematics well rather than to lament about
them not understanding a lot of mathematics (Andrews, 2001).
The recommendations by CBMS (2001) reconfirm that:
As
content courses are designed, it should be kept in mind that teachers will not
learn all the mathematics they need to know in their undergraduate studies---even
if, from their instructors' perspectives, the course content has been covered.
But if their undergraduate studies cultivate an interest in and capacity for
mathematical activity, teachers will be prepared to continue learning in the
context of their everyday practice. Furthermore, having developed in their
undergraduate training a curiosity about mathematical ideas and an appreciation
of mathematical pursuits, many more practicing teachers will be interested in
continuing their mathematics studies---an interest these institutions
should be prepared to address.
And
finally, the future teachers will need to be familiar and comfortable with a
variety of technological tools, including ordinary calculators, graphing
calculators, and computers, as well as appropriate geometric and computational
software. It will take time, effort, and money from mathematics departments to
incorporate the same into mathematics courses (Hungerford, 1994,
p. 18).
While
the above-mentioned reforms need a plenty of time and major changes at various
levels, there is a lot that individual instructors can begin to do for their
students immediately and make a difference. For example, as Thomas W. Hungerford (1994) suggests in his article, the teacher educators
can make their students more aware of the mathematical resources available to
them, particularly journal articles at their level. Most of the students are
not aware of the vast amount of mathematical literature designed for classroom
teachers. If possible, the teachers can have their students attend a meeting of
the state or local affiliate of the NCTM. This can be an eye-opening experience
for the students as they discover how much is going on in elementary school
mathematics. The teachers can also try to encourage group discussions for a
part of each class to model non-traditional teaching for the future educators.
And of course, the teachers can talk to their friends in various departments
about some innovative ideas as to how to make the lectures interesting even for
basic topics like addition algorithms (p. 19-20).
In
a project undertaken in
At
the
NCTM
has also taken some initiatives in the direction of improving elementary
teachers’ preparedness. As a start, this fall, the council launched its Academy
for Professional Development, which is offering two-day- and later will offer
five day- institutes focused on helping teachers put Principles and Standards
into practice (Stiff, 2001). The Achieve Foundations for
Success also tries to shed light on the preparation required at the elementary
level by presenting many problems and their solutions for middle school
students. A close survey of the problems that bring together various parts of
math like algebra and geometry reflects that the Achieve standards are
understanding-driven and expect students to reach a certain level of
understanding during elementary education more than expecting them to have covered
a certain range of topics.
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