# Teaching through Problem Solving

November 13, 2009 5 Comments

Problem solving is not only the reason for teaching and learning mathematics. It is also the means for learning it.

In the words of Hiebert et al:

Students should be allowed to make the subject problematic. … Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students. (Hiebert, et al, 1996, p. 12)

For years now, UP NISMED in-service training programs for teachers have organized mathematics lessons for teachers using the strategy we call Teaching *through* Problem Solving (TtPS). This teaching strategy had also been tried by teachers in their classes and the results far outweighed the disadvantages anticipated by the teachers.

Teaching through problem solving provides context for reviewing previously learned concepts and linking it to the new concepts to be learned. It provides context for students to experience working with the new concepts before they are formally defined and manipulated procedurally, thus making definitions and procedures meaningful to them.

**What are the characteristics of a TtPS?**

- main learning activity is problem solving
- concepts are learned in the context of solving a problem
- students think about math ideas without having the ideas pre-explained
- students solve problems without the teacher showing a solution to a similar problem first

**What is the typical lesson sequence organized around TtPS?**

- A problem is posed to the class.
- Students collaboratively solve the problem in many ways. (Role of teacher is to encourage pupils to try many possible solutions with minimum hints)
- Students studies/evaluates solutions. (Teacher ask learners questions like “Which solutions do you like most? Why?”)
- Teacher asks questions to facilitate connections among concepts, construction of new knowledge.
- Teacher extends problem.

**What are the theoretical underpinnings of TtPS strategy?**

- Constructivism
- Vygotsky’s Zone of Proximal Development (ZPD)

Click here for sample lesson using Teaching through Problem Solving to teach the tangent ratio/function.

A fun addition to this, I have found, is to get the class to solve a mastermind game as a group. Cracking the code involves a reasonable amount of logical thinking and playing it as a group encourages people to learn from each other.

Pingback: Misunderstanding of Understanding by Design | Keeping Mathematics Simple

Pingback: Teaching the properties of equality through problem solving « keeping mathematics simple

Pingback: Introducing the concept of function « keeping math simple

Phillips Exeter Academy has their whole math curriculum designed around a problem-based system. I have adopted/adapted this for my calculus and geometry classes.