Discussion:
Students must learn the precise use of the terms , operations , and symbols of
mathematics .
The ability to be precise is one of the great strengths (and requirements) of
mathematics
and the proper use of the language of mathematics is essential in this learning
process.
Mathematics is precise. The meanings of terms, operations, and symbols must be
completely unambiguous or communication is lost and mathematics slips away.
Students who are unsure of what they are talking about cannot hope to solve
problems with such ambiguous underpinnings. For students and teachers to
communicate about mathematics they must all have precise meanings for symbols
and terms in common. This is easy to overlook, and often overlooked, in
situations
where the content seems elementary, but this is exactly when that precision
should
start.
In order for terms and symbols to be precise they must all have definitions. As
much as arithmetic and algebra begin the content basis of mathematics, the
fundamental principles of mathematics revolve around explicit definitions of
terms
and symbols. There are two distinct aspects to the precision of mathematics.
Precision is necessary in the doing of mathematics and in the communication of
mathematics.
Abstraction is part of the underlying power of mathematics
and it should be taught from
the earliest grades. For example, physical manipulatives are teaching aids that
can help
lead students to understandings in mathematics. They are not, in and of
themselves,
mathematics, but are teaching tools to help get to the heart of mathematics.
One of the most important attributes of mathematics, and one that gives it much
of
its power, is its abstract nature. One piece of problem solving is the skill of
turning
complex word or story problems into concrete mathematics, in other words, to
abstract the mathematics from the problem. This allows the core mathematics to
be
understood separately from the application, i.e. in the abstract. The benefits
are
manifold. The same piece of abstract mathematics can be used over and over again
to solve a myriad of seemingly very different problems. Thus, an understanding
of
the abstract mathematics, together with the critical thinking skills necessary
to
extract a mathematics problem from a word problem, allows a student to solve
many problems with a small skill set. Without this abstraction each new problem
is
exactly that: a new problem requiring special mathematics just for it. For
example,
it is generally possible to figure out how to multiply any two given numbers
even
without the standard algorithm, but the problem is different for every two given
numbers. By abstracting the process to the standard algorithm for multiplication
we solve the problem of multiplication for all cases once and for all.
Another example is college calculus. The departments, such as physics, economics
and the various engineering departments, that require calculus for their majors
have
different applications in mind. The standard way to teach calculus for such
diverse
student needs is to abstract the material and teach the core principles that
apply to
all situations.
The place value system , subtract -fractions.html">fractions and the standard algorithms all contribute
greatly to
algebra readiness. In addition , students can learn to use variables and solve
pre-algebra
problems. They must also learn about graphs and graphing as functions.
Elementary school mathematics leads up to algebra and, as such, it should
prepare
students for algebra. The content of our five building blocks is all necessary
prerequisite for algebra, but, more than that, it prepares students for algebra
because there is a natural incremental transition from arithmetic to algebra. As
already mentioned, the place value system and skill (and understanding) with
manipulating the algorithms and fractions is invaluable preparation. More can be
done. Using abstract variables whenever possible and using commutativity,
associativity, and distributivity with the variables is great preparation.
Reading and drawing graphs are an important component of elementary school
mathematics. Using grade appropriate, but precise, definitions for functions,
students should learn to think of functions as rules , not just the formulas that
give
the rules. This can prepare a student for algebra by avoiding confusion when
functions are more carefully defined in algebra. It can also get students used
to
working with formulas and equations.
Textbooks can be a tremendous help to students. With a textbook, students have
the
opportunity to relearn and rethink what they have seen in class. Parents can
also have the
opportunity to use texts to help a student at home.
Textbooks allow students to look up concepts they are unclear on, learn from
reading (a skill frequently not attained by many college students) and rereading
(for
most, an essential ingredient while learning mathematics). The opportunity for
self
paced repetition comes with a text. Likewise, parents, friends, older students,
and
tutors can all help with the education process if the content is made explicit
in a
textbook. Not having a good text available is an unnecessary and severe handicap
placed on students.
Some students will not acquire all of the skills and understanding necessary to
begin a
formal study of algebra when the time comes. If they do not succeed at learning
a
fundamental third grade concept in the third grade they will struggle, and often
fail, to
learn more advanced concepts necessary in later grades. Such students are often
completely unprepared for algebra. These students need not be lost to the
subject. They
are older and more mature and should be given the chance to fill in their
knowledge gaps
through some review or remediation process.
For students to proceed on to algebra without the necessary background
mathematics is probably pointless. If they are given appropriate attention in
the
class, the prepared students will suffer. If not, the unprepared students will
suffer.
Intervention and remediation are the appropriate responses. Given the
opportunity
to take the time to memorize the single digit number facts, nail down the place
value system, learn to add, subtract, multiply and divide , such students have
the
potential to go on to learn algebra.
Certainly in high school there are two kinds of students, those who are still in
the
pipe line for a college mathematics course and those who are not. Pipeline
(college
preparation) algebra should not try to accommodate students without the
prerequisites. A placement test is probably appropriate for pipeline algebra to
be
sure students meet these prerequisites. This is very much for their good. If
they
are allowed to take algebra and then go to college without the necessary
arithmetic
background, they will take a placement test in college and find themselves in
remedial mathematics. (The majority of students who take a placement test in
college fail it.) When such students find they are missing their mathematical
foundation they tend not to be happy. They also seldom recover mathematically
well enough to proceed with a college level mathematics course. Students who
cannot place into pipeline algebra should have the option of proceeding with
non-pipeline
high school mathematics courses or getting the remediation that allows
them to proceed with algebra. Every opportunity should be made to allow students
to rejoin the pipeline. The vast majority of high school graduates go on to
college
and the best new jobs require some mathematics.
Final Comments.
Mathematics is not a collection of unrelated topics. Mathematics is
hierarchical. All of
the topics discussed fit into an interconnected dependency pattern. These
dependencies
form a structure: the structure of mathematics. This structure is really part of
the content
of mathematics and was built by mathematical reasoning and it takes mathematical
reasoning to make sense of the structure. The very structure of mathematics
dictates that
certain topics must be taught before others, and, in most cases, must be
mastered before
any understanding of the next topic can even be hoped for.
The core content for students who are in the pipeline for college mathematics is
not in
doubt. College mathematics teachers who teach students who need these courses
know
what is necessary. The key here is that ALL students in elementary school must
be
considered to be in the pipeline. Some college (or a college degree) has become
a
prerequisite for many of the new jobs being created in the United States. This
core
content for elementary school is actually quite small. For example, the top
performing
country internationally is Singapore. Their sixth grade textbook has a total of
less than
40 pages of instruction and examples. The rest is problems. Core elementary
school
mathematics content is straightforward and focused. The catch is that it must be
learned
well to progress.
Research base
In the reading debate questions were asked like: Is it better to teach reading
with phonics
or without phonics? Research could answer that question. On the surface, it
looks like a
similar question for mathematics is: Is it better to teach mathematics with the
place value
system or without the place value system? However, this question makes no sense.
The
place value system is mathematics! You cannot teach mathematics without the
place
value system, standard algorithms and our other building blocks.
The appropriate research questions are: How is the place value system best
taught?
What constitutes an adequate understanding of the place value system? What is
the least
painful way to acquire instant recall of the single digit number facts without
compromising understanding? What is the most effective way to teach the long
division
algorithm? Fractions are notoriously difficult for students. What approach, or
collection
of approaches, to fractions is most likely to succeed with almost all students?
New concepts in mathematics are presented to students each year of K-12
mathematics
education. How to best teach these concepts should be thoroughly researched.
Again,
this differs significantly from reading where getting started is what matters
most.
Postscript: Do we want domestically educated engineers?
A very high percentage of the U.S. professional science, technology,
engineering, and
mathematics personnel are foreign born and were given their K-12 mathematics
education in their home country. If we want homegrown engineers, certain things
have to
take place in our K-12 mathematics education system.
If students arrive at college with large gaps in their science education they
can survive,
college will essentially start from scratch with science, however undesirable
that may be.
This is not the case with mathematics. The concepts and skills developed in
every year
of K-12 mathematics education are essential to success in college mathematics,
mathematics that engineering students must all take. Manipulative skills with
numbers
and rational functions have been disparaged recently in education circles .
However, the
engineering student will face one class after another, year after year, where
the professor
comes in and writes equations on a blackboard for 50 minutes straight. Those
manipulative skills must be second nature in order to survive an engineering
course of
instruction.
Those necessary skills and concepts for the engineering student begin with the
foundation
discussed in this paper in early elementary school. There is a tendency to
suggest that
most students do not need all of these skills because most students will not
become
engineers. Even if this were true, and many believe that all students actually
need these
skills and concepts even if they are not going to be engineers, we would be in a
serious
quandary. Would this mean that we should not teach them to all students?
Students who
don’t get these skills and concepts will definitely not become engineers. So, if
we want
some students to be able to be engineers we have to teach these skills and
concepts to
these students. Is there any way to decide who in the fourth grade should be
given the
mathematics that would allow them to grow up to have the option of becoming an
engineer? Any attempt to separate elementary school children into two groups,
one
group that will never have the option of becoming an engineer and another group
that
will be given that option, would seem grossly unfair. All elementary school
children
should have the option of choosing to try to be an engineer, so all children
must be given
the necessary mathematics in elementary school.
