Posted in Mathematics education

What is scaffolding in education?

Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students learning. Some educational paper lists some of these scaffolding like “breaking the task into smaller, more manageable parts; using ‘think alouds’, or verbalizing thinking processes when completing a task; cooperative learning, which promotes teamwork and dialogue among peers; concrete prompts, questioning; coaching; cue cards or modeling”. Visual scaffolding is also popular in teaching mathematics.

Scaffolding is the latest buzzword in education community. In an international conference I attended recently for instance, I heard the word in almost all the parallel paper presentations.

There was a demonstration lesson for teaching English during the conference. I am not an English teacher so I asked the person seated beside me, who happens to be an English teacher, to tell me what the teacher was doing as she hopped from one group of students to the other. She said with authority that the teacher was doing a lot of scaffolding. I didn’t know what to make of her statement. Was it a positive or a negative comment? Is it a good idea to do a lot of scaffolding or is it something that should be given sparingly? Where do you draw the line?

Scaffolding can be traced back to Lev Vygotsky’s idea of ZONE of PROXIMAL DEVELOPMENT (ZPD). Vygotsky suggests that there are two parts of learner’s developmental level: 1) the Actual developmental level 2) the Potential developmental level

The ZPD is “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance”. This is where scaffolding is crucial.

scaffolding in education
Note that the activity students should be engaging in is problem solving. A problem is a problem only when you do not how to solve it right away. So when scaffolding deprives the students from thinking and working on their own way of solving the problem then scaffolding has not helped learn how to solve problem. It only helped them to solve problems using the teacher’s method.

You may want to read the different interpretations of zone of proximal development in research.

Posted in Curriculum Reform, Mathematics education

Teaching through Problem Solving

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