Exploring How Meanings
are Made in Mathematics
Minnie Loh
Maryknoll Convent School
Topic: Introduction to Probability [Form 3]
3 Elements of Mathematical Language
• John Polias (2007) stated that teachers
can make meaning in Mathematics
through three elements. They are
• (1) Linguistic
• (2) Visual
• (3) Symbolic
(1) Linguistic
• The linguistic includes the language of the
mathematical text and the language used by
teachers and students in dealing with
mathematical knowledge.
(2) Visual
• Visual representation includes diagrams,
graphs and pictures. They are used to make
meaning in Mathematics.
(3) Symbolic
• Makes mathematical meaning through
symbols and the arrangement of those
symbols.
Topic (field):
F.3 Mathematics – Introduction to Probability
Main objectives of the unit of work:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Explore the meaning of probability
Calculate the theoretical probability by listing
the sample space and counting
Investigate geometric probability
Compare experimental and theoretical
probability
Find the expected number of occurrences
of an event
Find the expected value of an event
Lesson 1 & 2
(i) Explore the meaning of probability
<Linguistic>
–
–
–
–
Concept of the likelihood of an event happening.
A number can indicate more clearly the likelihood
or chance for an event to happen.
Introduce the terms ‘probability’, ‘sample space’,
‘impossible event’ and ‘certain event’
Concept of equally likely outcomes through
examples.
<Symbolic>
P(E) 
frequency occurences favourable E
.
total frequency possible outcomes
Lesson 3 & 4
(ii)
Calculate the theoretical probability by
listing the sample space and counting
<Linguistic>
- Introduce the term ‘theoretical probability’
<Symbolic>
- 0  P( E )  1
- P( E )  P( E ' )  1.
- Tree diagram & tabulation method
to list all the possible outcomes
Lesson 5
(iii) Investigate geometric probability
<Linguistic>
- Introduce the term ‘geometric probability’
<Visual>
- Calculate the probability by referring to the areas or
length of the figures involved.
Lesson 6 & 7
(iv) Compare experimental and theoretical
probability
<Linguistic>
- Introduce the term ‘experimental probability’
<Visual>
- Perform an experiment by throwing a fair dice
Lesson 8
(v) Find the expected number of
occurrences of an event.
<Linguistic>
- Introduce the ‘expected number of
occurrences of an event’.
<Symbolic>
- Calculate the expected number of
occurrences of an event through examples.
Lesson 9 & 10
(vi) Find the expected value of an event
<Linguistic>
- Introduce ‘expected value of a trial’
<Symbolic>
- Introduce the calculation of expected value of an
event through examples.
Remarks about the teaching strategy:
Introduce the Mathematics definitions
to students
by unpacking nominal groups.
Pattern for definitions:
Word being defined + relating process + class or generalization
[Nominal group]
[embedded clause or phrase]
Example:
A ratio is an ordered comparison of quantities of the same kind.
This definition pattern can help students to see how
definitions are constructed through unpacking the
nominal group.
Examples used in this topic:
Example:
The number used to indicate the likelihood of an event happening is
called its probability.
Probability is the number used to indicate the likelihood of an event happening.
Example:
The collection of all possible outcomes is called the sample space.
The sample space is the collection of all possible outcomes.
Examples used in this topic: (cont’)
Example:
If the probability of an event happening in a single trial is p, then
we expect after n trials this event will occur np times.
Suppose p is probability of an event happening in a single trial.
The expected number of occurrences of an event after n trials
is np times.
Advantages of Unpacking Nominal Group
• Catch students’ attention by identifying the
word being defined first.
• Build up students’ understanding by
showing the class that the word belongs to.
• Finally, qualify elements are used to
elaborate the term to strengthen students’
understanding.
In fact, the technique of unpacking nominal
groups in mathematics is not just for
understanding definitions.
This can also help students to identify the
nominal groups in the math problems, so that
students are clear about the meaning being
made in the question.
Example:
There are 120 apples in box. Given that the probability that
an apple is rotten is 0.08, find the expected number of
rotten apples in 100 boxes of apples.
Recall:
Suppose p is probability of an event happening in a single trial.
The expected number of occurrences of an event after n trials is np times.
0.08 is probability that an apple is rotten.
The expected number of rotten apples in a box of 120 apples is 1200.08 = 9.6.
 The expected number of rotten apples in 100 boxes of apples
= 9.6100
= 960
Thank You