Mathematics in Context
Where is it going
Mathematics in context
The study of mathematics year after year may seem endless and pointless
for many students and teachers.
The key elements in high school mathematics prepare for calculus.
Calculus is the subject which is used to justify calculations and
formulas in mathematics and other subjects. It requires an efficient
command of arithmetic without a calculator. It also requires in full
logic and the algebraic way of writing and reasoning. Most elements of
geometry with and without coordinates, and trigonometry too with
triangles and unit circles are required in calculus.
The connection of decimals, ratios, rates, percentages to each other and
fractions is part of arithmetic. Mastery of Statistics (means, averages,
pie charges) exercises skills in arithmetic and knowledge with
fractions. While calculators are useful, the ability to work efficiently
with fractions with no calculator present is required for algebra or the
shorthand roles of letters and symbols in describing calculations and
showing how to obtain them. probability provides an opportunity to
review and reinforce fraction sense and efficient operation with
fractions - must for algebra.
Most other elements of high school can be presented in ways that support
and strengthen the key elements. Dilatations, if taught provides practice
with algebra and arithmetic operations, and may be used to imply
similarity of diagrams in radial expansions, contractions and inversions
about a point or origin. Spatial sense or three dimensional geometry in
secondary can be linked to crystal shapes, to technical drawing or
drafting and to arithmetic and algebra via volume and surface formulas,
and via Euler relation between the vertices, faces and edges of polygons
in plane.
Statistics in the lower high school curriculum provides exercises with
arithmetic, the proper and improper gathering and interpretation of data
and associated graphs. Recognizing what is improper is a base for
critical thinking when numbers and graphs are presented. The prerequisite
and context for statistical critical thinking is a proper and efficient
command of arithmetic without a calculator and familiarity with
computations with methods or formulas lead to repeatable and reproducible
results. The latter leads the notion that numbers do not lie, statistics
might. But there are some statistical methods for which may be present
only because they are required for course final examination - not an
inspiring reason for mastering them.
Mastery of high school mathematics with its emphasis on arithmetic
without a calculator (we hope) and the algebraic way of writing and
reasoning provides a key to studies in the physical sciences, physic and
chemistry included. The difference between the advance and ordinary
student of the phyical science comes in part from mathematical skills.
The advance course requires a mastery of fractions, what they are and
operations with them, exact solution methods for linear equations in one
to two unknowns, and mastery of the quadratic formula. Anything less
implies difficulty for the student.
Engaging and Inspiring Teachers
The prerequisite to an engaged student is an engaged, inspired and
informed teacher.
The foregoing may provide a context for high school mathematics – lines
of reason that tie the various elements together. The fact that five
sixths of high school students in secondary IV do not have the arithmetic
skills or fraction sense named a prerequisite for algebra in objectives
for the first year of high school points to a difficulty. The above
context engages me and therefore provides a framework for engaging
students in the curriculum, so that it becomes a more meaningful sequence
of skills and concepts.
Engaging Students
One way to engage student could be to point out the appearance of
fractions and calculations in their daily life to give them more
participation and ownership of the lesson.
More generally to improve study skills and work habits would be to start
this is a guided discussion with students of how one learns or masters a
subject, discipline or collection of skills and knowledge in general. The
discussion may span lessons and separate times. A consensus on what they
expect from the teacher and how to favour those expectations by
cooperating in their own education is a goal. For example, cooking,
swimming, body-building (like it or not), nordic skiing, bicycling,
drawing, writing, figuring (do arithmetic without a calculator) are
examples which students could discuss to provide them insights into the
learning and teaching process. Here the question for students to
considers is how they would teach or communicate a skill or concept, an
isolated one, or a sequence. The question of how to teach another an
organized set of skills or an organized body of notion may lead to
students to the model of skill and idea verification and development, one
skill or idea, one skill after another. That expectation gives a base for
learning mathematics and support for the statement of goals and standards
to guide and focus their work.
The site before you was developed to support direct instruction, the case
where the teacher explicitly controls the direction of a class and what
is in it. That may lead to a teacher or subject centered classroom. The
student-centered classroom calls for more students participation, more
realistic and more authentic examples and projects to engage the
student's interest and cooperation in his or her learning. The
instructor in the latter case has to provide or limit the examples and
projects to the broad or narrow subject or discipline or them under
construction. The notion that a student can rediscover key skills and
concepts is false. Those key skills and concepts are the product of many
investigations and of thought on how to arrange the findings into
coherent islands and bodies of knowledge. None the less, some combination
of teacher led and student led activities which allow the student to
beyond the passive reception of ideas and concepts is called for. How is
another question which ideologues for pedagogical methods expect the
instructor to construct. Direct instruction requires less preparation
than instructional methods which call for student participation in that
direct instruction only requires content mastery in a discipline and
lines of reason and skills to present and verify. I would recommend
teacher training colleges in the first instance focus on content mastery
to set a base for the more complicated or more involved student centered
approach.
Carrot and Stick
The workplace usually has a firm organization based on the meeting the
requirements of the employer by coercion, do this or else be fired - lose
your income. the school classroom in many English speaking lands do not
have a firm organization. Nominally, instruction is for the sake of the
student. However, education is often justified by the need for for better
educated population for the sake of a competitive or suffering economy.
For students, compulsory education for 10 years or so may be followed by
the uncertainty of the workplace: will the student be employed or not?
will education help? Education would be more authentic or appealing if
students saw favorable consequences, rewards or reasons for studying.
Education for its own sake may not be appealing to a student and the
student's family without explicit authentic and realistic advantages Not
enough hope dims enthusiasm for education.
The young student goes to school with enthusiasm, a curiosity and a will
to grow-up (so I imagine or recall). But belief in education is somewhat
like belief in Santa-Claus. It may be transient. In the end some
students or most serve time in high school, suspecting it may help in a
general way, but not being exactly sure how. Authentic consequences to
education would help prolong student enthusiasm and patience with being
educated.
More notes - Personal
As a mathematics instructor, I enter a classroom where most students who
would take another mathematics course but for mathematics being required
for a diploma or for future studies. I have to motivate them and provide
a context for the task ahead, namely skill and concept verification and
development reaching for the goals and standards. The first section above
mathematics in context gives a context for my instruction. The
call for inclusion of authentic, realistic, cross-discipline examples,
activities and themes cannot be criticized save for the demand that I the
teacher to find and invent ways to comply. The callers do not provide a
manual. At the same time, mastery of my subject calls for students to
see definitions and explanations, and do exercises to confirm and
reinforce those definitions and explanations and to build general
thinking or general problem solving skills in lines of reasoning that
define the discipline. Just as the body builder does exercises with no
immediate consequences, so must the mind-builder. I am looking for ways
to engage students to give them the goals and standards, and hence will
to follow a sequence of exercises, listening to definitions and
explanations included, whose net result in a repeatable and reproducible
fashion for the patient at least is mastery of mathematics and its
logic. The next time I teach I hope to maintain a written record via
checklist or rubrics of the abilities of each of my students, so that in
dealing with each, I know at glance where the weaknesses and strengths
lie. Individualized or differentiated instruction may follow, time
permitting, or as needed.
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