Tag: teaching thinking via mathematics

Posted in What is mathematics

Math is not easy to learn – that is a fact.

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I think it’s a waste of time trying to make math easy and fun to learn if your idea of fun does not involve challenge.

Mathematics is not an easy subject and it is not easy to learn it. That is a fact. The sooner the teacher accepts this, the better for her students. The challenge to us teachers is not in how we can make math easy to learn but in how we can make it makes sense and how we can make our students love the challenge that mathematics presents. Can math be challenging if students feel that what they are expected to do in the class is to follow the teacher’s method, the teacher’s way of thinking, and the teacher’s way of doing things? Where is the fun in that?

Mathematics is not fun to learn if the idea of fun is like playing bingo! However, if ‘fun’ is a function of the challenge a sport or a game presents, then indeed learning mathematics is fun. We love a sport because of the challenge it presents, the opportunities it gives us to make prediction, analyze, strategize, make our stand and defend it, etc and not because it is easy to play!

Everything in mathematics makes sense. Everything in mathematics is connected to everything else. I think this is where we teachers should be devoting our time to. And this is what this blog is about!

Posted in Curriculum Reform, Mathematics education

Teaching through Problem Solving

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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?

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

What is the typical lesson sequence organized around TtPS?

  1. An which can be solved in many ways is posed to the class.
  2. Students initially work on the problem on their own then join a group to share their solutions and find other ways of solving the problem. (Role of teacher is to encourage pupils to try many possible solutions with minimum hints)
  3. Students studies/evaluates solutions. (Teacher ask learners questions like “Which solutions do you like most? Why?”)
  4. Teacher asks questions to help students make connections among concepts
  5. Teacher/students extend the problem.

What are the theoretical underpinnings of TtPS strategy?

  1. Constructivism
  2. Vygotsky’s Zone of Proximal Development (ZPD)

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

The best resource for improving one’s problem solving skills is still these books by George Polya.

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I