## Guiding Questions to Check for Math Understanding

*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 at the beginning of the lesson, I feel that we are telling them what the present is before they open it. I like to keep the suspense of what students learn by allowing them to discover or uncover the key concepts of the lesson for themselves. This approach is fundamental in inquiry-based learning models. How do we draw out conceptual understandings from our students without 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 from asking what the formula for the circumference of a circle is. The formula C = **π**d, 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, and 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 questions:

- 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 from 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 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 of work I would recommend teachers design guiding questions to help students understand the synergistic relationship between the factual content and the conceptual content in a unit of work.