How do we give our students a present without telling them what it is before they open it?

Often when we tell our students what they are going to learn and understand at the beginning of the lesson I feel that we are telling them what the present is before they open it. Another analogy my colleague at HKU uses is give students a trailer to the film not the spoiler! It is important to set the stage the learning but not spoil the learning! I like to encourage curiosity and excitement with my students by allowing them to discover or uncover the key concepts of the lesson for themselves. This approach is fundamental to inquiry based learning models.

So how do we draw out conceptual understandings from our students with telling them?

Guiding questions are a vehicle to draw out conceptual understandings (generalizations) from our students. Guiding questions are also known in education circles as essential questions and are a critical part of unit planning.

What are some examples of guiding questions? If we were studying a unit on circle geometry and we wanted our students to understand that the ratio of circumference to diameter of all and any circle presents a fixed constant, π

Some guiding questions could be:

What is the definition of circumference?

What is the definition of diameter?”

What is the formula for the circumference of a circle?

Other questions to elicit more thought and understanding could be:

How do you describe the relationship between the circumference and diameter of any circle?

This type of question is different to asking what the formula for the circumference of a circle. The formula C = πd, in symbols, does not reflect a statement of understanding however if we ask students to describe the relationship between circumference of diameter for all and any circles this reflects a deeper conceptual understanding of the concepts of fixed ratios, diameters, circumference.

Three Categories of Guiding Questions: Factual, Conceptual and Debatable/ Provocative.

Factual Questions are the “what” questions such as “what is the formula for the circumference of a circle”. Factual questions ask for definitions, formulae in symbolic form and memorised vocabulary. Often factual questions begin with the start: what is…?

Here are some more examples of factual:

What is y=mx+b?

What do the letters m and b stand for?

What is the quadratic formula?

What do the letters a,b and c stand for in the quadratic formula?

Conceptual questions use the factual content in a unit of work as a foundation to ask students for evidence of conceptual understanding. Often conceptual questions start with: how or why…?

Here are some more examples of conceptual questions:

How is a variable different to a parameter?

How does the concept of “mapping” explain the concept of a function?

How would you describe direct proportionality between two variables mean?

How does y = mx+b represent a translation and a transformation?

How do we model real life situations using functions?

Debatable or provocative questions create further curiosity and debate and provoke a deeper level of thinking. For example for a unit on circles a debatable or provocative question could be: Is a circle a polygon?

Other examples of debatable/ provocative questions are:

Were logarithms invented or discovered?

How well does a linear function fit all situations in real life?

How reliable are predictions when using models?

When planning a unit, I would recommend teachers design guiding questions that specifically align with statements of conceptual understanding to help students understand the synergistic relationship between the factual content and the conceptual content when learning math.