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.