Active learning of mathematics
Abstract (summary)
Smith
discusses the importance of active
learning techniques in
the mathematics classroom.
A teacher working with this view would have to provide students with
opportunities to create their own theories and to engage their mental
model-making processes.
Full Text
SINCE THE
COCKCROFT REPORT IN 1982,
THERE HAS BEEN AN increasing emphasis on the use of active learning in school mathematics, with a typical and influential view
being propounded in Better Mathematics: "Mathematics can be effectively
learned only by involving pupils in
experimenting, questioning, reflecting, discovering, inventing and discussing. Mathematics should be a kind of learning which requires a minimum
of factual knowledge and a great deal of experience in dealing with situations using particular kinds of
thinking skills" (Ahmed 1987, 24).
During the
same period of time, there has been an increased explicit and implicit use of a
constructivist epistemology, for example in an implicit way: "The teachers job is to organize and
provide the sorts of experience which enable pupils to construct and develop
their own understanding of mathematics,
rather than simply communicate the ways in which they themselves understand the subject" (NCC
1989, para. 2.2).
And in an explicit way: "Many
writers embed their view of active
learning in a framework concerned with the nature of the intellectual
activity taking place, most often located within a constructivist model of
mathematical learning"
(Kyriacou 1992, 312).
We can, with
some consistency, summarize this particular constructivist framework as the
hypothesis that human knowledge is personally constructed and consists of
conjecture, unfalsified theories, modified theories and expectations.
A teacher
working with this view of how we come to know would have to provide pupils with
appropriate chances to create their own theories, to engage their mental
model-making processes and to allow them the opportunity to develop
expectations in order to
subject their theories to the test of reality. This [approach] strongly
suggests that pupils must be actively engaged in constructing their understanding, and that the
activities themselves must be judged mainly by their contribution in assisting pupils to construct
their own understanding of concepts selected by the teacher.
Within this
particular constructivist framework, learning
tasks should be chosen with the specific intention of actively involving
learners in seeking to
understand their external world by creating and testing their own models of
what is going on "out there" in reality. The teacher has a clear initial role in selecting activities that are
expected to focus the attention of the learner on constructing the intended learning outcomes of the session.
The mathematical activities should therefore be selected or designed to
encourage the learner to link between external world and internal thought. This
involves a consideration of presentation, pupil activity, reflection and
socialization. Activity by itself is not enough.
Presentation
WE MUST AIM
TO PRESENT ANY MATHEMATICAL ACTIVITY in
a way that invites pupils to fully engage their higher mental capacities. This
can be by the use of a game, a puzzle, a surprise or some other intriguing
challenge. In creating a
challenge for learners, we must be aware of the need to choose an appropriate
level of challenge; one that learners can perceive as offering them a
realistic, but not certain, chance of meeting. The challenge can come from the
teacher or from the pupil themselves. It can be in the form of a puzzle, a target to reach, a goal
or a conflict to resolve.
An example
of a presentation that involves all four of these aspects is to motivate
pupils' work on geometrical construction. One way in which I have done this is to show pupils a
hexaflexagon, on the front of which is drawn a pattern and on the reverse, a
different pattern. Pupils are asked to memorize the patterns, and whilst they
attempt to do this, I flex the hexaflexagon whilst maintaining the same pattern
on the front, then ask, "Who can remember the pattern on the back?"
and surprise the class by showing that the pattern has "disappeared."
The goal is then set for the class members to make their own hexaflexagon,
resolving the puzzling disappearance of the lost pattern. I have found this to
be a simple yet highly motivating presentation, which sometimes generates a round
of applause-I wish I could say that more often!
In creating the initial challenge, surprise and cognitive conflict can be
useful to generate interest. For example, by asking pupils to work in pairs and giving one of each
pair a basic calculator and the other a scientific calculator, we can create
cognitive conflict and surprise if we ask pupils to carry out calculations
(like 2 + 3 x 5) that give different answers on the two calculators. The role
of the presentation is to indicate to the learner the need for new or revised
theories; the role of the challenge, surprise or cognitive conflict is to
engage the learners' full attention.
Pupil
Activity
TO SOME
EXTENT THE ROLE OF PUPIL ACTIVITY IS clear; it is for learners to undertake in order to meet the challenge
set by themselves, the teacher or the text. They should be trying to make sense
of the challenge, the activity and their findings. It can therefore be to move
toward the goal, or to attempt to resolve a cognitive conflict, or even to
explore the extent of the cognitive conflict. For example, to continue with the
calculator challenge, we might ask pupils to find out as many sums as they can
that lead to different answers on the different calculators.
The key
aspect of any learning
activity is that it must be constructed or chosen to demand mental involvement;
perhaps this is best achieved by requiring pupils to deal with new, unfamiliar
and nonroutine activity. Familiar mathematics
can often be packaged in an
unfamiliar way; for example, I recently worked with a Y10 top set class
studying graphical inequalities. To do this, I began by placing the pupils in a rectangular array and
systematically giving each person a set of coordinates. By asking pupils to
stand up if they satisfied inequalities like 2x + 3y > 5, I involved all the
pupils in thinking about
the familiar coordinates in
an unfamiliar way. The class teacher reported that "the activity
maintained interest, was fun and entertaining and got across what I usually
find pupils have difficulty with very clearly. The physical nature of the task
helps keep interest and minds awake!"
The task
should not be a passive routine, such as factorizing fifty similar-looking
equations. Practice may be important to develop skill, but from this
perspective the less routine there is in
it, the more learning is
likely to be achieved. In
writing of the importance of a firm conceptual understanding, HMI state that
". . . progress in
pupils' mathematical understanding is more important than progress in the performance of skills. In fact, when the early stages of
learning are firmly
established subsequent progress can take place more quickly and
confidently" (HMI 1985, 36).
Reflection
ON
COMPLETION OF AN ACTIVITY, A REVIEW OF THE learning achieved during the activity can be most helpful in assisting the learners to
integrate their new or revised theories and expectations with their other
mental systems.
"Activity
per se is not a guarantee of mathematical learning" (Goodchild 1992, 24). Goodchild goes on to
consider the importance of reflection and "interpretation," which he
sees as a mechanism for making sense of the learning activity and for locating it in a wider framework of meaning
and purpose. Alternatively, it may be seen as a process in which new personal theories are created or
existing personal theories modified and in which new expectations may be created. The teacher's role in this process is to ensure that
there is time for such reflection, and to provide a mechanism to ensure that
reflection occurs. The classroom organization of reflection may involve pupils in writing, or it may involve
structured discussion with other pupils or with the teacher.
"When
asked for the connections between practical work and the symbolic statement of
rule, the children's best reply was that one was a quicker route to the answer
than the other. Nobody mentioned that the practical experience provided the
data on which the formula was built. The teachers did not stress why this
procedure was being followed, nor emphasize the generalizability of the rule
and thus the advantage of accepting it" (Hart 1989, 139).
Within the
theoretical framework outlined above, it may be seen as appropriate to ask
pupils to be explicit about some new expectations. This obliges pupils to
create both theories and expectations. To continue the calculator example: once
pupils have amassed a number of calculations that give different answers on two
calculators, they can be asked to predict the two answers for other
calculations, to predict when the calculators will give the same answers and to
predict when they will give different answers.
Socialization
"KNOWLEDGE,
FROM THE CONSTRUCTIVIST POINT OF view, is always contextual and never separated
from the subject. . . to know also implies understanding in such a way that the knowledge
can be shared with others and a community thus formed. A fundamental role is
played by the negotiation of meaning in
this interaction, which is of a social nature" (Moreno-Armella and Waldegg
1993, 657).
I do not
believe that many pupils achieve this socialization of knowledge whilst working
from individualized schemes. Even the pronunciation of key mathematical words
(e.g., "sin" instead of sine) is missing, let alone the opportunity
to discuss mathematical problem solving in depth. Socialization cannot be delegated to textbooks.
Having
developed an understanding of a mathematical concept, there is a need for
pupils to be able to communicate effectively about it. For this [requirement],
there is often a need to know, understand and use the appropriate language and
terminology and to adopt the standard conventions. In other words, there is a need to socialize the
personal and private understanding. The teacher has a role here in ensuring that the need for
communication is apparent, that activities incorporate discussion work to
facilitate such communication and that pupils are helped to become aware of
conventional language and notations. "The use of discussion as a technique
for teaching centres on the
fact that it is above all else a means of escaping from our own individual
perceptions of the world, with all their circumstances and boundaries into
which we would otherwise be locked. It adds to the richness of understanding
and enables us to make contact with the minds of others in the most direct way possible" (Van
Ments 1990, 17). During the socialization process, the individual is being
expected to negotiate a shared meaning with the teacher, peers, external
examiners and textbooks largely by coming to grips with conventions and
conventional language. If all pupils in
a group are encouraged to communicate about their mathematical activity, this
[approach] can provide a richer learning
environment for each individual, as well as begin the process of negotiating
shared meanings and socializing the new knowledge.
Conclusion
IN THIS ARTICLE, I HAVE CONSIDERED A CONSTRUCTIVIST framework for the
analysis of mathematical learning
activities. This analysis suggests that mathematical activities are not enough
to achieve learning by themselves;
they need to carried out with a consideration of aspects of presentation, the
nature of the pupils' mental activity, the need to ensure pupil reflection and
the achievement of socialization of the learning.
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