Mathematical investigation refers to the sustained exploration of open-ended mathematical situations, as opposed to problem solving which has a definite goal. The aim of mathematical investigation is to develop students' mathematical habits of mind through processes like systematically exploring situations, formulating problems and conjectures, and justifying conclusions. Mathematical modelling applies mathematics to real-world problems in order to understand the situation, and may use empirical, simulation, deterministic, or stochastic approaches. Problem posing refers to generating new mathematical problems or reformulating existing ones, and develops higher-order thinking skills.

1. Mathematical
Investigation and
Modelling
Samantha Nicole Gelizon
BSED 3- Mathematics
• Closed Versus Open-Ended Problems
• Problem Posing

2. What is Mathematical Investigation?
• Mathematical investigation refers to the sustained
exploration of a mathematical situation.
• It distinguishes itself from problem solving because it is
open-ended.
• Problem solving is a convergent activity. It has definite
goal, the solution of the problem. Mathematical
investigation on the other hand is more of a divergent
activity.

3. The ultimate aim of
mathematical investigation
• Develop students’ mathematical habits of mind.
Mathematical Processes which are the emphasis of
mathematical investigation:
• systematic exploration of the given situation
• formulating problems and conjectures
• attempting to provide mathematical justifications for the
conjectures

4. What is Mathematical Modelling?
• The process of applying mathematics to a real world
problem with a view of understanding the latter.
• Mathematical modeling is the same as applying
mathematics where we also start with a real world
problem.
• The modeling process may or may not result to
solving the problem entirely but it will shed light to the
situation under investigation.

5. Four Broad Approaches of
Mathematical Modelling
• Empirical Models
• Simulation Models
• Deterministic Models
• Stochastic Models.
*The first three models can very much is integrated in
teaching high school mathematics. The last will need a little
stretching.

6. Empirical Modelling
• Involves examining data related to the problem with a
view of formulating or constructing a mathematical
relationship between the variables in the problem using
the available data.
Simulation Modelling
• Simulation modeling involves the use of a computer
program or some technological tool to generate a scenario
based on a set of rules. These rules arise from an
interpretation of how a certain process is supposed to
evolve or progress.

7. Deterministic Modelling
• Deterministic modelling in general involves the use of
equation or set of equations to model or predicts the
outcome of an event or the value of a quantity.
Stochastic Modelling
• Stochastic modeling takes deterministic modeling one
further step. In stochastic models, randomness and
probabilities of events happening are taken into account
when the equations are formulated. The reason behind
this is the fact that events take place with some
probability rather than with certainty. This kind of
modeling is very popular in business and marketing.

8. Closed-Ended Problems
What is a closed question/task?
• Closed tasks ask students for one answer and there is
usually only one way to get the correct answer.
Example:
"What is 3+4?," or "Do you understand what a
rectangle is?"

9. Open-Ended Problems
• Often named “ill-structured” problems as they involve a
higher degree of ambiguity and may allow for several
correct solutions.
Example:
“How much water can our school save on a period of
four months?” or “Design a better gym room
considering the amount of money we can spend.”

10. FEATURES of open-ended problems:
• There is no fixed answer (many possible answers)
• Solved in different ways and on
different levels (accessible to mixed abilities)
• Empower students to make their own
mathematical decisions and make room for own
mathematical thinking
• Develop reasoning and communication skills

11. Problem Posing
According to some authors…
• Problem posing refers to the generation of new
mathematical problems and the reformulation of given
ones (Chapman, 2012). It requires higher order thinking
skills and has become a recognized means of developing
mathematical thinking and creativity in students of all
ages (Koichu & Kontorovich, 2013).

12. On the other hand…
• Problem Posing means creating problems from the
content.
Why it is Important?
• In this process the students will gain knowledge and
required confidence.

13. Thank you!