*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.

## Fred / September 29, 2016

ReRead “The Dialogues” by Plato. Socrates said the same thing some 1400 years ago and he was right that this the way to mathematics, although it is difficult in a classroom setting. His conclusion that all learning was essentially “remembering” things already in the mind, he was right about learning mathematics in that it is all logical extensions of things we know.

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## Jennifer Wathall / October 1, 2016

Hi Fred,

I agree that the classroom setting may present challenges using guiding questions to draw knowledge and understandings from our students. Hopefully this structure of using 1) Factual 2) Conceptual 3) Provocative questions will support this process more. Thanks for sharing!

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## Jennifer Wathall / September 27, 2016

Hi Asesh,

Traditional classrooms have focused on rote memorization of skills and processes rather than the underlying concepts of mathematics. Mathematics is a language of conceptual relationships and using guiding questions is a strategy to draw these understandings from our students. Another important part of unit planning is the crafting of generalizations. “Concept-Based Mathematics: Teaching for Deep Understanding in Secondary Schools” describes in detail how to create and plan a concept-based mathematics curriculum.

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## Asesh Datta / September 27, 2016

Agree with your thoughts. How do we inculcate mathematics concepts to begin with? In our development, mathematics is an understanding of numbers and its variants. This is definitely different from literature or mother tongue. Love of a subject comes from these conceptual understanding. The figure of pi as a constant must be appreciated. Why this constant is unique? Story based math teaching will make the subject enjoyable.

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