Making Connections through
Promoting Mathematical Thinking
Lim-Teo Suat Khoh, MME, NIE
Mathematics Teachers Conference
2 June 2011
Overview
Mathematical Thinking and
Connections in the Singapore
Mathematics Curriculum
Connections within
mathematics strands
Connections across strands
Connections with the real world
Mathematical Thinking and Making Connections
Aims of Mathematics Education in Schools
Develop the mathematical thinking and
problem solving skills and apply these skills
to formulate and solve problems.
Recognise and use connections among
mathematical ideas, and between
mathematics and other disciplines.
Do these sound familiar?
Mathematical Thinking and Making Connections
Students should use various thinking skills and
heuristics to help them solve mathematical
problems. Thinking skills are skills that can be used
in a thinking process, such as classifying,
comparing, sequencing, analysing parts and wholes,
identifying patterns and relationships, induction,
deduction and spatial visualisation.
Connections refer to the ability to see and make
linkages among mathematical ideas, between
mathematics and other subjects, and between
mathematics and everyday life. This helps students
make sense of what they learn in mathematics.
Do these sound familiar?
Singapore mathematics curriculum
The Singapore mathematics curriculum
promotes making connections and thinking
skills.
Think for 30 sec and tell your neighbour
which mathematical thinking skill or skills
you most often encourage in your classes.
Then tell him/her what motivates you to
encourage those thinking skills or what
value you see in encouraging those
mathematical thinking skills.
Mathematical thinking skills are the essence
of mathematics. A person does
mathematics when he/she engages in such
thinking processes.
Mathematical thinking skills provide
connections which makes mathematics
topics meaningful – otherwise we just have
a repertoire of disconnected facts and rules.
Topics in the Syllabus
Number Algebra
Space
Data
Topic 1
Topic 1
Topic 1
Topic 1
Topic 2
Topic 2
Topic 2
Topic 3
Where are the connections between topics
within or across strands?
Connections within Strand
Making Connections Within Strand
As Mathematics is largely hierarchical, it is
necessary to build new concept on those
previously established and learned.
Making such connections are essential as they
give justification for the new concept learned.
Euclidean Geometry as a field of study is the
natural strand to develop connections since the
whole structure of concepts, theorems and
properties are connected via logical reasoning
bridges.
In topics of other strands, building of one
concept upon another is also necessary.
Example 1: Constructions
Compass constructions of angle bisector
and perpendicular bisector of line segment
Think through the steps of the construction
Do you simply tell your students the steps?
Do they know why the steps work?
Example 1
Symmetry properties of
isosceles triangles
A Kite is made up
of two isosceles
triangles with
equal bases
Line of symmetry is the
perpendicular bisector of
the base and is the angle
bisector of the third
(unequal) angle.
Construction of
perpendicular bisector
of line segment
Properties of kites
Diagonals are perpendicular,
one of the diagonals is the
angle bisector of two angles
and it also bisects other
diagonal perpendicularly
From a diagonal,
Constructing
constructing a
a kite and its
kite and
diagonal
symmetry
diagonal
Construction of
angle bisector
Lim, S.K., (1997), Compass Constructions: A vehicle for promoting relational
understanding and higher order thinking skills. The Mathematics Educator, 2(2)
Example 1: Reasoning and Deduction
Use of questioning to encourage reasoning
is good but be careful of over-scaffolding
which may be counter-productive to getting
students to reason for themselves.
Hierarchical teaching requires planning the
order of the topics.
If curriculum topics across years are not in
the order wanted, the connection can be
made later but within the same year,
teachers can plan the ordering of the topics
with such linkages in mind.
Example 2: Trigonometry – solution of triangles
How are the following topics linked?
Pythagoras’ theorem
Sine/cosine rule and solution of triangles
Write down Pythagoras’ theorem and the
cosine rule.
c 2 = a2 + b 2
c2 = a2 + b2 – 2ab cos C
A
C
A’
B
Example 2
Generalisation and Specialisation
Learning of the special case of right-angled
triangles first
Moving towards more general case of any
triangle – as angle C is reduced to acute
from right angle, the side c is shortened.
Pythagoras’ Theorem is used in proof of
cosine rule
Pythagoras’ Theorem becomes a special
case of cosine rule where  C becomes a
right angle and cos C = 0
Structures and Relationships
In mathematics, defining concepts,
making categories and sub-categories
and establishing relationships is a
fundamental process.
This uses thinking processes like
comparing, contrasting, generalising,
exemplifying.
The strength of mathematics is that
rules/processes work for a category rather
than just for particular cases.
Understanding this enables efficient
processes to be carried out.
Example 3: Categories of Quadrilaterals
Properties of quadrilaterals are learned in Sec 1 but
students seem only to learn them as separate shapes
each with its own properties.
It is difficult to remember so many properties
Are they able to see that
Every property satisfied by a parallelogram is also
satisfied by a rectangle or a rhombus
Every property satisfied by a rhombus is satisfied by a
square
Every property satisfied by a rectangle is satisfied by
a square
How can we teach the topic so that the relationship
between the shapes are understood?
MCK Question from TEDS-M
Three students have drawn the following Venn diagrams
showing the relationships between four quadrilaterals:
Rectangles (RE), Parallelograms (PA), Rhombuses
(RH), and Squares (SQ).
Which student’s diagram is correct?
Among NIE’s graduating student teachers who are
trained to teach primary mathematics, only 66%
obtained the correct answer.
Categorising, Specialising, Exemplifying
The concept of set relationships promotes the
mathematical understanding that an element of a
subset satisfies every property that an element of
the superset satisfies.
Thus, a square which satisfies every property of a
rhombus and every property of a rectangle is both
a rhombus and a rectangle. The set of squares is
thus the intersection of the set of rhombuses and
the set of rectangles.
Categorising, Specialising, Exemplifying
Activities that can lead to such understanding
includes getting students to try to create impossible
shapes after studying the properties e.g. give me
an example of a square which is not a rhombus.
Such activities also encourage higher van Hiele
level thinking – understanding the meaning of a
mathematical definition. Misconceptions from
primary levels can also prevail – a rectangle must
have 2 longer sides and 2 shorter ones.
A square is a rhombus because it satisfies all
the properties which define a rhombus.
Example 4: Categories of numbers
Why are there different types of real
numbers: natural numbers, negative
numbers, integers, rational numbers?
What is the relationship between them?
Organisation of subsets of real numbers
Set of Real Numbers
Negative Numbers
Positive Numbers
Z Counting
Integers E Numbers
R
O
Rational Numbers
Expansion of the concept of numbers: from counting numbers, to fractions, to
negative numbers and irrationals.
Subtraction and Closure
The concept of closure for operations.
The set of whole numbers is NOT closed
under subtraction
Need for
negative
numbers !
Expansion of concept of numbers
Thus within the set of whole numbers (most
of primary school arithmetic), there needs to
be constraints on subtraction (subtract
smaller number from bigger) and division
(remainders) in order for “answers” to still
be whole numbers.
In order for subtraction to be closed, we
need to expand to negative numbers.
Moral of Xuan’s story – even primary children
can understand concept of negative numbers
Expansion of concept of numbers
Closure of operations
Set
Natural Numbers
Integers
Rational Numbers
Real Numbers
+



















For division to be closed, we need to expand to
rational numbers – unfortunately, most students do
not see fractions as answers to whole number
division.
Connections (asking appropriate questions, providing
challenging examples) help students to see the
relationship between the types of numbers.
Categories of Numbers
As example of disconnectedness, many pupils
do not see fractions as a type of numbers.
Are whole numbers fractions?
Why are operations different? Or are they?
Number line is a useful tool to see fractions as
numbers – this is one concept of fractions not
well established due to over emphasis on
“part of a whole” concept of fraction over other
concepts.
What are the links between these concepts?
Addition of fractions
1 4 3 8 11
   
2 3 6 6 6
4
3
0
1
2
1
11
6
If I choose to add 2 and 3 by writing 2 as 6/3
and 3 as 12/4 and use the rule for adding
fractions would the answer be correct?
2
Example 5: Polynomial Functions
Polynomials include linear functions, quadratic
functions, cubic functions.
Our pupils learn from particular to general,
beginning from linear, to quadratic to
polynomial but unless we then connect them
up to those learned earlier, each becomes a
separate category disconnected with those
learned earlier.
Importance of contrast and compare to
“recognise” a particular concept i.e. necessity
of non-examples, variation in examples and
superordination.
Example 5
When teaching quadratic expressions
ax2 + bx + c:
Get students to give a few quadratic expressions
– try to have more diversity in the coefficients
(negative, non-integers)
Ask them to change their quadratic expressions
slightly so that they are no longer quadratic
When teaching polynomials, students can be asked
to give examples of polynomials they have already
encountered.
Always include non-examples, tricky examples etc
when establishing concepts
Example 6: Using processes learned earlier
In teaching processes, it is also important to
make connections so that students know we
are using processes learnt earlier.
Substitution is a very powerful tool in
mathematics because it allows the person
transform what he is working with into a form
he can work with because of processes
learned earlier.
So here the skill is the altering process and
the connection is to a solution method already
mastered.
Example 6
Solving simultaneous equations
y2 + (2x + 3)2 = 10
2x + y = 1
Solving equations where unknown is in the
exponent
e-x(2e-x + 1) = 15
Integration
Differentiation (chain rule)
Summary of learning theories
Principle of
Variability
Multimodal
Contrast
(nonexamples)
Higher level
concepts
Teach
-ing
Building on
schema
Piaget, Bruner, Dienes, Gagne, Skemp, Marton
Concept Mapping
Use of topic maps across
years could be useful for
teachers
Concept maps within a
topic cluster can provide
learner with overview of
what (s)he has learned
and what the connections
are.
Preferably, students can
draw their own maps after
each topic cluster.
Functions
Average
rates of
change
Graphs
Intuitive
understanding
of tangent as
instantaneous
rates of change
Derivative
Limits
Connections across strands
Across domains of Number, Algebra
Algebra as generalised arithmetic,
to provide a language to articulate
rule
Letters as pattern generaliser
Algebra as processes to solve
problems
Letters as unknowns
Sec 1 teachers to link algebra to
model method
Number
Across domains Algebra and Geometry
Algebra as a language to articulate
relationships
Letters as variables
Cartesian geometry expresses relationships
in terms of points in space across dimensions
Use of Algebraic processes to solve
equations in geometry, trigonometry –
meaning of solution as values that satisfy
the relationship which is expressed as one or
a set of equations.
Across Data, Algebra and “Geometry”
Algebraic language used to express
relationships from data
Visual representation to illustrate data
relationships in a different mode e.g. scatter
plots.
Connections to Reality
The applicability of Mathematics
Usefulness of basic mathematics is obvious
for maths at primary level but not secondary.
Applications are at higher levels beyond
what the secondary students see and they
regard attempts to apply as impractical,
unnecessary or irrelevant to their lives.
May need to go for novelty effect or link to
what they are interested in e.g. sports, how
points are awarded in games and how these
are linked to strategies, mathematics in
nature.
Going broader – Points to watch out for
Reality check – complexity of real life, nonlinearity of relationships, information
gathering, extraneous information, different
perspectives.
Using data to make informed choices,
decision making skills from multiple
perspectives should be encouraged.
Letting students choose their own problems,
which could depend on the school context,
latest fashion.
Work with teachers of other subjects for
cross-discipline projects.
Going deeper – beyond syllabus
Tickle their mathematical fancy – go further,
beyond curriculum demands e.g. a result
has been established in a 2-dimensional
plane; what about in 3-dimensions or on a
curved surface?
Challenge them to use deeper mathematics
e.g. Calculus for growth models.
Start with a problem and allow students to
find the mathematics necessary (which they
may not have learned yet or may never
learn)
Thinking skills fostered/encouraged
Hypothesising
Checking hypothesis
Making inferences
Explaining
Convincing/justifying (not just mathematical
proofs)
Giving examples
Making generalisations
Conclusion
Content-wise
Connections are
the essence of
mathematical
structures
Pedagogy-wise
Connections
enhance learning
(better and
deeper learning)