The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in.

###### Standard exercises

These are activities with clearly defined procedure/strategy and goal. Standard exercises are used for mastery of a newly learned skill – computational, use of an instrument, and even new terms or vocabulary. These are important learning activities but must be used in moderation. If our teaching is dominated by these activities, students will begin to think mathematics is about learning facts and procedures only. This is very dangerous.

###### Problem solving activity

These are activities involving clearly defined goals but the solutions or strategies are not readily apparent. The student makes decision on the latter. If the students already know how to solve the problem then it is no longer a problem. It is an exercise. Click here for features of good problem solving tasks. It is said that problem solving is at the heart of mathematics. Can you imagine mathematics without problem solving?

###### Math investigations

These are activities that involve exploration of open-ended mathematical situation. The student is free to choose what aspects of the situation he or she would like to do and how to do it. The students pose their own problem to solve and extend it to a directions they want to pursue. In this activity, students experience how mathematicians work and how to conduct a mathematical research. I know there are some parents and teachers who don’t like math investigation. Here are some few reason why we need to let our students to go through it.

- Students develop questions, approaches, and results, that are, at least for them, original products
- Students use the same general methods used by research mathematicians. They work through cycles of data-gathering, visualization, abstraction, conjecturing and proof.
- It gives students the opportunity communicate mathematically: describing their thinking, writing definitions and conjectures, using symbols, justifying their conclusions, and writing and reading mathematics.
- When the research involves a class or group, it becomes a ‘community of mathematicians’ sharing and building on each other’s questions, conjectures and theorems.

Students need to be exposed to all these type of mathematical activities. It is unfortunate that textbooks and many mathematics classes are dominated by exercises rather than problem solving and investigations tasks, creating the misconception that mathematics is about mastering skills and following procedures and not a way of thinking and communicating.

Samples of these tasks are shown in the picture below:

if juan cut a144-inch-long piece of wood into 8-inch pieces, how many pieces will he have. this is a promble-sloving investigation: math. choose a strategy

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