Enhancing critical thinking dispositions in the mathematics classroom
~ Magda Kloppers ~
A positive disposition towards mathematics (and all other subjects) should be developed by students
and teachers need to help their learners to do so. The National Curriculum Framework as well as the
Curriculum and Assessment Policy Statement promote effective learning by mastering critical thinking
skills and dispositions, inter alia by successfully identifying and solving problems and collecting,
organising and evaluating information. Critical thinking dispositions in mathematics (and other subjects)
refer to the ability to search for alternative solutions to problems (to be open-minded), to follow
reasons and evidence (truth-seeking), a systematic approach and inquisitiveness (to be curious). The
enhancement of critical thinking dispositions should be included in curricula, tasks and assessments to
produce students who are able to think critically. Classroom settings in which students structure their
own knowledge and in which they develop confidence in their skills to use mathematics (language,
science etc. ) should be created. Learners sometimes give up, they do not persist, are passive and wait
for the teacher to give an answer. Perseverance can be enhanced by making use of the Elements of a
Plan as a strategy. Different questions should be asked throughout this cycle to help learners to persist
but also to encourage dispositions like open-mindedness, truth-seeking, curiosity and systematicity. To
achieve these goals, I found the Elements of a plan as structured by Feuerstein and Polya, a useful
strategy.
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Learners will be given a copy of the elements of a plan to refer back to constantly.
In the following explanation, an example from mathematics will be used to
demonstrate how critical thinking dispositions can be encouraged in classroom
settings. Define the goal
When planning a lesson, the educator should,
apart from the curriculum outcomes also
formulate outcomes that should be achieved in
terms of critical thinking dispositions. An
example might be:
At the end of this study section, learners
should understand the value and meaning of
open-mindedness,
systematicity
and
curiosity; or
Learners should listen attentively to their
peers’ views on ….; or
Learners should be able to use a variety of
strategies in solving problems and in
communicating the solutions.
The academic goal can be to teach simple and
compound interest in financial mathematics:
If one deposits R 500 into a bank account at 6%
simple interest, calculated annually, what will the
amount in the bank be after four years?
When the outcome is to teach learners to
use a systematic approach, a real-life
example and questions can be given.
“What does a supermarket do to make it
easier for the public to find articles?”
(Articles are sorted on shelves according to
their uses) and “How does this relate to
Mathematics?” There is a systematic path
that one follows in solving a problem. We
do not just “jump in”. One needs to read
carefully through the question and analyse
it. The meaning of a systematic approach
in mathematics is that there is an order in
which calculations are performed. The
value is to be accurate and clear, and one
has to be sure that the assignment is
completed successfully. Questions should
be asked throughout the lesson to
encourage learners to think. The abovementioned questions may also motivate
learners to learn new things about a wide
range of topics and to be well informed
(curiosity/inquisitiveness). Time should be
allowed for learners to be tolerant of
divergent views (open-minded).
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What do I have?
What strategy will I use?
Where shall I start?
When referring to the above example, the Where do we find symbols? What symbols
following questions might be asked:
would you find in a car, the kitchen, money
market. What does these symbols mean?
What are Symbols?
(inquisitiveness, open-mindedness, truthseeking)
In our question, what does that mean?
In financial Mathematics we find: A, P, r, n
Why are these values (interest rate for
loans and for savings) different? How does
What is the current interest rate? If you owe, it affect one?
(truth-seeking/openinvest money?
mindedness).
Which one would you
prefer? When learners are given the
opportunity to reason, they need to be
What does n represent?
tolerant and should respect the rights of
others to have different opinions (openWhat is the difference between linear (simple minded).
interest) and exponential (Compound interest)?
Do we also find n ’s in nature, on our
calendar?
Where else do we find symbols?
By asking questions learners are encouraged to think critically and to be open-minded. They are
also encouraged to work systematically and to communicate their ideas clearly. This can also
lead to confidence in their own reasoning. Curiosity is also evoked and learners can apply what
they do to real-life situations.
What do I have to calculate? A, P, r or n?
By asking these questions, emphasis is on
the order of calculations (systematicity),
How will I do this? Using a formula or pencil and and also that one should make sure about
paper - reasoning it out without making use of a the most effective way to solve a problem
formula?
(truth-seeking).
Why is a formula better?
What other formulas does one find?
Identify the variables and values given and group
them.
Substitute values into the formula.
Students should link these questions also to
science, geography and other disciplines
(curiosity).
As students tend to give up, they should
have a plan, a method according to how
they will work.
By explaining this
important step in the Elements of a Plan,
students should be able to express their
thinking in words and then do the
mathematical part (systematicity).
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In mathematics these are seldom given. One is to
find the correct answer. It is expected of students
to know and apply all the other unstated “rules”.
They should be able to:
Substitute values into the formula.
What are the rules?
Work systematically.
Get an exact answer.
Multiply the values.
Remember their units.
Although this is a simple example, more
advanced calculations will follow (this is
only an introduction) and therefore
learners must be made aware that there is
a sequence/order in which calculations
must be done – especially when getting to
compound interest. Accuracy is important.
Units are also very important in
mathematics and students should be aware
of this.
Solve the problem
I  Pr t
 500  0.06  4
 R 120
and then ... reflect:
Does this answer appear to be realistic?
The question was not to find the interest, but the
amount in the account after 4 years. This answer
is therefore not complete. The initial amount
should be added.
Is there another way in which we could do the Students need to evaluate their answers.
same calculation?
Open-mindedness and truth-seeking is
encouraged.
A  P (1  rt )
Reflect
 500 (1  0.06  4)
 500 (1.24)
 R 620
Learners should continually refer back to the Elements of a Plan as this is a strategy to help them to
persevere and not to leave any question unanswered. I always tell my students that if a problem seems
to be very difficult they should indicate answers to the questions of the Elements of a Plan next to the
problem. In doing so, fewer problems are left unsolved.
Magda Kloppers is a senior lecturer in Mathematics in the School of Education Sciences on the Vaal
Triangle campus of the North West University in Vanderbijlpark. Prior to this position, she was teaching
Information Technology, Computer Applications Technology and Mathematics in high schools. She
holds BSc, PGCE, MEd and PhD qualifications.
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