Transcript Slide 1

Using design to enrich mental
constructs of a mathematical
concept
Dr Zingiswa MM Jojo
Department of Mathematics Education
University of South Africa
Introduction
• Instructional design is the process by which
instruction is improved through the analysis of
learning needs and systematic development of
learning material (Carter, 2011)
• creation of effective meaningful lessons
• helping students to make sense of information
• cut through extraneous information
The choice of instruction to be used in a lesson
depends on:
The design of instruction and activities
• the teacher’s knowledge of the concept,
• preconceptions,
• misconceptions and
• the difficulties that learners could experience in learning
the concept.
To understand a particular mathematics concept or topic
involves knowing the relationship between various topics
and where a particular topic fits in the bigger picture
Curriculum Shifts
• Learner centred approaches and arguments
have replaced examination focused, teachercentred, and content driven approaches for deep
understanding of mathematical concepts
• These bear evidence of teachers’ struggle in
putting them to practice in their classrooms in
South Africa due to both cultural and resource
contexts.
Understanding a mathematical concept
Wiggins:
(1) explanation,
(2) interpretation,
(3) contextual applications,
(4) perspective,
(5) empathy and
(6) self-knowledge.
In order to think about mathematical ideas there
is a need to represent them internally in a way
that allows the mind to operate on them
As relationships are constructed between internal
representations of ideas, they produce networks
which could be structured like vertical hierarchies
or webs.
With regard to learning mathematics with
understanding:
• a mathematical idea or procedure or fact is
understood if it is part of an internal network
• the mathematics is understood if its mental
representation is part of a network of
representations.
•
The degree of understanding is determined by
the number and strength of connections.
Initial genetic decomposition (IGD)
IGD refers to the set of mental constructs which
the learners should construct in order to
understand a given mathematics concept.
The genetic decomposition was composed in
terms of mental constructions (actions, processes,
objects, schemas) and mechanisms (contrast,
separation, generalisation and fusion) learners
might employ when learning inequalities.
APOS and Variation
Variation Interaction
 Variation interaction is a strategy to interact with
mathematics learning environment in order to bring
about discernment of mathematical structure.
 Variation is about what changes, what stays
constant and the underlying rule that is discerned by
learners in the process.
 Instructional design to enrich mental constructs from
the lens of variation and how learners are brought to
the schema level of understanding a concept.
Variation
Based on Leung (2012) ‘the object of learning’ i.e. What is to be
learnt? –discernment of what is to be learnt in the lesson
• Contrast presupposes that for one to know what a
concept is, he/she has to discern and know what it is not.
e.g. examples and non-examples
• Separation assumes that all concepts have a multitude
of features, each of which give rise to different
understandings of the concept.
• Generalisation refers to the verification and conjecture
making activity that checks out the validity of a
separation.
Variation continued
Fusion
is
the
simultaneous
discernment of all the critical features
of a concept and a relationship
between them which allows a learner
to make connections gained in past
and present interactions
Research Question
• What is the nature of instructional
support that can generate in students
the kinds of mental representations
that will enable them to think about
these critical differences when
engaging in symbol manipulation
activity involving inequalities?
Questions to consider for instruction
• What are students’ conceptions of
inequalities?
• What is typical correct and incorrect
reasoning?
• What are common errors?
• What are possible sources of students’
incorrect solutions?
• What are promising ways to teach the
topics of inequalities?
Challenges for Teaching the topic
Inequalities are:
• taught in secondary school as a subordinate subject (in
relationship with equations),
• dealt with in a purely algorithmic manner,
• taught in a manner to avoid the difficulties inherent in the
concept of function.
• taught in a sequence of routine procedures, which are
not easy for students to understand, interpret and control
• algebraic transformations are performed without taking
care of the constraints deriving from the fact that the >
sign does not behave like the = sign
Processes- Action
interiorisationtransformations in
the mind
Separationdiscernment of the
critical characteristics
of a concept to
differentiate it from
others
Generalisationverification and
conjecture making
activity on the
separated pattern
Contrast- to know and
discern
what
a
concept is and what it
is not
Actions- Physical
manipulations external
to the mind
Objects-process
encapsulated-
FusionTotality
of
actions
Schema- reflection on
process as a totality of
knowledge
An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to
“x ≥ -2”
means that whatever value x
has, it must be greater than
or equal to -2.
Try to name ten numbers that
are greater than or equal to
-2!
Numbers greater than -2 are to the
right of -5 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
Solve an Inequality
w+5<8
We will use the same steps that we did with
equations, if a number is added to the variable, we
add the opposite sign to both sides:
w + 5 + (-5) < 8 + (-5)
All numbers less
than 3 are
solutions to this
problem!
w+0<3
w<3
More Examples
1 -2y ≤ 5
1 - 2y + (-1) ≤ 5 + (-1)
-2y ≤ 4
All numbers from -2 up (including -2)
make this problem true!
Example
Teacher 2 - Concept Map- IGD
Contexualising the problem
• The doctor instructed my
grandfather to take no more than
3 pills per day
• My instructor advised me to run at
least 10km per day to prepare for
the marathon
• I ate at most two meals today
Example “x < 6”
Teacher 2:What is meant by this?
Are those the only numbers?
To the answers given, the teacher probed by
suggesting non-examples
How can we represent this on the number line?
Opened up the scope of knowledge- encouraging
critical thinking
The learner has to convince himself- Internalize
the knowledge
Discussion- Teacher 1
Teacher 2
• Contextualised the problem- using an authentic
task as an instructional strategy
• make sense of some of the exceptional
transformation rules used in solving inequalities
• properties underlying valid equation-solving
transformations are not the same as those
underlying valid inequality-solving
transformations
• multiplying both sides by the same number,
which produces equivalent equations, can lead
to pitfalls for inequalities
Conclusions and recommendations
• Discernment of the concept - the learner
to be aware of certain features which are
critical to the intended way of seeing this
concept
• Highlight the essential features of the
concepts through varying the nonessential features
• Give the learners a chance to
demonstrate argue and explain their
solutions to others
• See learners as constructors of meaning
• help learners to actively try things out,
• Experience the construction of multiple
perspectives of mathematical concepts,
• Find components of the concept that are
interconnected with each other
• extend the original problem by varying the
conditions, changing the results and
generalize;
• (2) multiple methods of solving a problem by
varying the different processes of solving a
problem and associating different methods
of solving a problem
• (3) multiple applications of a method by
applying the same method to a group of
similar problems.