Lesson 10: Mathematical Investigation

there would be given a certain number of children. Some may want to find out how ~iany high-fives there would be if instead of once, the children would high-five each :th e r twice or thrice. Some children may even decide to work on a problem that the teacher has not thought of. This is investigation as an activity itself.

As illustrated, what sets mathematical investigation apart from other strategies that have been discussed in this unit by far is that the goal of the investigation is -ot specified by the teacher; the students have the freedom to choose any goal to oursue. In problem-solving, the students are encouraged to think outside the box; in •nathematical investigation, there is no box to start with. The students are placed in a space where they can play around whichever way they want. This makes mathematical investigation a divergent and learner-centered strategy. So, like in the problem-solving strategy, it is crucial that the teacher chooses or creates a situation that is engaging and caters mathematical investigation. Tasks A and B show that a close-ended word problem can easily be converted into an open-ended investigative task by simply replacing the question with an instruction to investigate.

There are three main phases of a mathematical investigation lesson: the (1) problem- posing, (2) conjecturing, and (3) justifying conjectures. In the problem-solving phase, the students explore the given situation and come up with a mathematical problem that they would want to engage in. The conjecturing phase involves collecting and organizing data, looking for patterns, inferencing, and generalizing. In the final phase, the students are to justify and explain their inferences and generalizations.

Always remember that although mathematical rules or theorems may arise as results of the mathematical investigation, they are not the objectives of an investigative lesson—the objective is the investigation itself; the exercise of creative thinking and problem-solving that the students underwent as they investigated. Mathematical investigation is not after the teaching and learning of some competency in the curriculum; it is about developing the mathematical habits of the mind.

The only planning that the teacher needs to do is to create or choose an appropriate task and anticipate possible problems that the students would pose. Below is an example of a close-ended word problem transformed into a mathematical investigative task and the problems that the students would possibly come up to.

Close-ended problem:

There are 24 animals in a farm. Some are cows and the rest are chicken. There are 60 animal legs in all. How many cows and how many chickens are there?

Investigative task:

There are 24 animals in a farm. Some are cows and the rest are chicken. Investigate.

Possible student-generated problems:

1. How many cows and how many chickens could there be?

2. What is the possible total number of animal legs?

3. Given any total number of animals, what is the ratio of the number of cow legs to the number of chicken legs?

Assess

The following activity will broaden your understanding of the mathematical investigation strategy.

1. Use the Venn diagram below to compare and contrast problem-solving and mathematical investigation.

Challenge

Even though the students are the ones who would identify the problem given a situation, the teacher must be able to anticipate some of the problems that may come up. To develop this skill, the teacher must undergo mathematical investigation.

1. Pose a problem, make a conjecture, and justify your conjecture given the following situation. This task is adapted from Orton and Frobisher's Insights into Teaching Mathematics (1996).

Investigate the following number tricks: 854 ~458 396 +693 1089

Harness

Choose a close-ended problem from the DepEd mathematics teaching materials for Grades 4 to 6. Transform it to an investigative task then list down the possible problems that the students could pose given the task. This activity will be part of the learning portfolio that you will compile at the end of this module.

Close-ended problem:

S u m m a r y

Mathematical investigation is an open-ended teaching strategy that capitalizes on the students' ability to identify a problem. Any word problem can be transformed in to a mathematical investigation by limiting the given information and omitting the specific question that it is asking.

O b j e c t i v e

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Execute the Empathize, Define, Ideate, Prototype, and Test stages of the design- thinking process

Introduction

The students find learning mathematics most engaging when they are involved in a thinking process that results in an output that can be applied to relevant context. The design-thinking process engages the students in such a thought-provoking and purposeful activity.

Design thinking is a progressive teaching strategy that allows the students to look for real-world problems and finding creative solutions. Students do this by focusing on the needs of others, collaborating for possible solutions, and prototyping and testing their creations. This can be summarized in five stages: Empathize, Define, Ideate, Prototype, and Test. These stages are adapted from the Institute of Design at Stanford University.

The goal of design thinking is for the students to respond to a particular need (a real-world problem) so it is fitting that the first stage is empathy. In this stage, the teacher needs to be explicit in guiding the students to put themselves in the shoes of others through activities like immersed observation and interviews. According to the

T h in k

Design Thinking Framework (Institute of Design at Stanford, 2016)

Empathize