INDEPENDENT
LEARNING AND MATHEMATICS
Nora Nineachtain,
ILC Coordinator, HCT, Dubai Men’s College
If
thou art diligent and wise, O stranger, compute the number of cattle of
the Sun, who once upon a time grazed on the fields of the Thrinacian isle
of Sicily (Archimedes)
The purpose of this article is to initiate an exchange of ideas on mathematics
within an independent learning (IL) framework. Historically, as far back
as the Babylonian civilization, mathematics learning was equated with problem
solving for everyday situations combined with the mental gymnastics of working
out real or fictitious puzzles. Such was the evolution of mathematics; a
fun and recreational activity, engaging the mind and fostering a community
of learners. As early as the third century BC, Archimedes was writing to
Eratosthenes
of Cyrene and to many of his students using epigrams to solve “The Cattle
of the Sun” puzzle cited above. This puzzle is just one of the many fascinating
mathematics challenges that have occupied scholars throughout the ages.
Early in the 20th century, H.E. Dudeney’s Amusements in Mathematics, followed
closely by Martin Gardner’s Mathematical Circus, will be well remembered
as bringing recreational mathematics to the attention of all. More recently,
Ian Stewart in Math Hysteria has attempted to keep such traditions going.
Are these authors familiar to all of us? Did you study such mathematics
books at school? Were your courses modelled on Euclidean methods of abstract
proof and the development of algorithms, the right answer being the only
goal? Is the excitement of mathematics, through exploration and discovery,
included in courses? All too often, the evolution of exciting and intriguing
number patterns such as the Fibonacci numbers, Pascal’s Triangle or Fractals
is neglected and only encountered by accident in publications such as The
Da Vinci Code.
Is the learning and teaching of mathematics on the way out? A recent article
in The Guardian (see below for link) reported on the closure of yet another
mathematics department in the UK – continuing an alarming trend.
So, is mathematics within an IL framework the answer?
What is REAL? … Real isn't how you are
made ... It`s a thing that happens to you … It doesn't happen all
at once ... You become ... It takes a long time ... (Margery Williams)
Unanimous consensus regarding interpretations of independent learning is
rare. Mathematics within an independent learning framework could be considered
to depend on: existing learning strategies; students’ knowledge to date;
wise use of e-learning technology; a 24/7 availability of expertise; a wide
range of self-access resources, student motivation; the cultural background
of students; previous learning experiences of the students involved; the
content; the “learning to learn” strategies integrated into the mathematics
learning activities; validated diagnostics; the availability of remediation
packs; student readiness for the mathematics area; contextualising all content
... the list is endless!
This article proposes that mathematics is perfectly poised to create engaging
learning experiences which would encourage the development of independent
learning strategies. Such experiences, reflected upon and integrated into
the students’ portfolio of IL skills, is the key to both independent learning
in mathematics and to the transfer of such strategies into the learning
of other subjects.
Does this mean that we abandon formal teaching,
no longer teach algorithms to solve problems, or hope that learning by “osmosis”
will happen once students are presented with the “appropriate” learning
activities? A holistic approach to creating mathematics learning
experiences through the integration of other subjects – nature, music, history,
art and English language (to name but a few) – could make learners more
involved in the process of learning. This kind of involvement can also be
enriched by off-line and on-line collaboration, and by interaction and engagement
with the course content. All too often, the freedom to experiment is lost
as course content is matched with unrealistic time-frames. The challenge
facing us today is to create mathematics learning environments and learning
cycles which will empower students to devise approaches to learning uniquely
suited to them, and which would equip students with ever evolving learning
strategies. As Margery Williams said, “...
it`s a thing that happens to you … it doesn't happen all at once ... You
become ... It takes a long time”.
As the purpose of this article is to initiate an exchange of ideas on mathematics
within an independent learning framework, what
questions should we be asking? What are your own views? What resources would
you recommend to encourage mathematics within an independent learning framework?
I would recommend highly the “Living Worksheets” from http://www.livingworksheets.co.uk/,
not just as stand-alone activities, but as a valuable resource to be used
in many ways. For mathematics problems, games and articles, try the NRICH
site at http://www.nrich.maths.org.uk/public/index.php.
If, on the one hand, it is expected that mathematics learning can be broken
down into a bite-sized dissemination of skills, then a mathematical wasteland,
as mentioned in The Guardian’s article, looms! On the other hand, in this
the 21st century, inherited pedagogical wisdom could partner technological
expertise to create dynamic learning experiences. Mathematics learning could
turn the full circle and revolve around explorations, discoveries and simulations.
For more information, see Margery Williams’ The Velveteen Rabbit (Doubleday and Company:
Garden City, New York). Inspiration can also be found in Dr Seuss’ Oh, the Places You'll Go!
For more information on H.E. Dudeney, see: http://www.kalva.demon.co.uk/dudeney.html
The Guardian report on the closure of mathematics departments can be found
at
http://education.guardian.co.uk/higher/news/story/0,,1411319,00.html