Using design to enrich mental constructs of a mathematical concept

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Using design to enrich mental constructs of a
mathematical concept

• Dr Zingiswa MM Jojo
• Department of Mathematics Education
• University of South Africa

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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

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The design of instruction and activities

• the teachers 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

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Curriculum Shifts

• Learner centred approaches and arguments have
  replaced examination focused, teacher-centred,
  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.

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Understanding a mathematical concept
Wiggins (1) explanation, (2)
interpretation, (3) contextual applications,
(4) perspective, (5) empathy and (6)
self-knowledge.
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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.
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• 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.

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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.

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APOS and Variation

•  

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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.

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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.

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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

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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?

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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?

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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 gt sign does not behave like the
  sign

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Separation-discernment of the critical
characteristics of a concept to differentiate it
from others
Processes- Action interiorisation-transformations
in the mind
Objects-process encapsulated-
Generalisation-verification and conjecture making
activity on the separated pattern
Contrast- to know and discern what a concept is
and what it is not  
Fusion-Totality of actions
Actions- Physical manipulations external to the
mind
Schema- reflection on process as a totality of
knowledge
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HOW and WHERE do I start with the teaching of
inequalities?
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An inequality is like an equation, but instead of
an equal sign () it has one of these signs lt
less than less than or equal to gt
greater than greater than or equal to
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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!

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Numbers greater than -2 are to the right of -5 on
the number line.

• 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.

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Solve an Inequality
w 5 lt 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) lt 8 (-5)
w 0 lt 3
All numbers less than 3 are solutions to this
problem!
w lt 3
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Do we do the same with the next example? When do
we deviate from this routine procedure?
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More Examples
1 -2y 5
1 - 2y (-1) 5 (-1)
-2y 4
 
All numbers from -2 up (including -2) make this
problem true!
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Example

•  

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Teacher 2 - Concept Map- IGD
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How does this help with the learners learning of
the concept?
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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

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Example x lt 6

• Teacher 2What 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

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Discussion- Teacher 1

•  

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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

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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 non-essential features
• Give the learners a chance to demonstrate argue
  and explain their solutions to others

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• 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

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• 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.

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Thank you Ndiyabulela