Posted in Assessment

Teach and assess for conceptual understanding

To teach for conceptual understanding requires assessing for conceptual understanding. The principles that guide teaching for understanding must be the same principles that should guide assessment. What are some of these?

  • Open-ended, problem solving tasks
    To teach for conceptual understanding, it is not enough that students engage in problem solving task. The tasks should be (1) open-ended which means that it can be solved in many ways using a range of concepts; (2) accessible, that is , not too easy or too difficult but just beyond the students ability; (3) can be extended by changing conditions in the problem so that it can be used for building concepts and for making synthesis and generalization; and, (4) the task should encourage creativity in the problem solver. These, together with right amount of scaffolding from the teacher and assessment tasks possessing the same characteristics is a perfect recipe for understanding mathematics conceptually.
  • Activities that promote mathematical communication
    Mathematics is a language that enables us to communicate ideas with conciseness, clarity and precision both in oral or written form. Students learning experiences should always aim at developing this capacity. They should be given opportunities to talk about mathematics, to speak mathematics, and communicate mathematically through its written symbols.  These are possible with the right mix of collaborative and individual work. Click this link for sample. This also implies that assessment should focus not only on the knowledge the students are acquiring but also on their skill on communicating this knowledge.
  • Tasks that build on students’ previous knowledge
    Teaching should build on the knowledge that students already have. This does not mean simply putting something on top of what they know. Knowledge has to be connected with other knowledge from within and from without. The more connections there are, the more robust is the understanding. Conducting formative assessment can provide teacher with information on how to structure the lesson to help students make connections. Another strategy which I highly recommend is to teach via problem solving. Click here for sample lesson.
  • Discussions that respects reason
    Mathematics is a way of thinking logically and methodically. As such, classroom culture that respects reason must be created both in the teaching and in assessing. Group or whole class discussion and assessment rubrics should give appropriate feedback to the students as to the way they reason and build on each others reasoning or on each others opinion.
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