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 = 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, 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.

CRAFTING QUALITY MATH GENERALIZATIONS: ESTABLISHING GOALS TO FOCUS LEARNING

One of the Mathematics Teaching Practices from National Council of Teachers of Mathematics (NCTM, 2014) is:

Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses goals to guide instructional decisions. 

Generalizations are summaries of thought and overriding ideas that we want our students to understand from their study. They are our mathematics goals to focus learning. Lynn Erickson (2007) first coined the term “generalizations” and in education circles generalizations are also known as enduring or essential understandings. It is a statement that connects two or more concepts in a sentence of relationship. Here are some examples of generalizations:

Students understand that: 

  • Linear functions show relationships that exhibit a constant rate of change
  • A function represents a mapping (rule) that assigns each input (domain) with one output (range)
  • For right-angled triangles, the area of the square drawn from the hypotenuse represents the sum of the areas of the squares drawn from the other sides
  • Logarithm laws provide a means of changing multiplicative processes into additive processes and can be used to find inverses of exponential functions

If my students said any of the above statements I would be satisfied that they have fully understood the topic and actually have a deep understanding of the concepts involved.

To help craft quality generalizations Erickson (2007)  suggests a scaffolding process.

Here is a specific example of how to scaffold a generalization in order to  improve the quality of the generalization.

Level 1 Students will understand that…..Logarithm laws affect logarithmic expressions
Level 2 How or why?Logarithm laws reduce logarithmic expressions utilizing  the inverse process of exponential functions which represent continuous compounded growth.
Level 3 So what? What is the significance or effect?Logarithm laws provide a means of changing multiplicative processes into additive processes and this can provide the means to find inverses of exponential functions (continuous compounded growth).

In summary here are the three levels for crafting generalizations:

Level 1 Students will understand that…
Level 2 How or why? (Choose which question is most appropriate)
Level 3 So what? What is the significance or effect?

It is important to note here that critical understanding in mathematics is usually achieved through a level 2 generalization therefore the target for curriculum development and instruction is level 2 type of generalizations. Level 3 generalizations are used if we wish to extend an idea or apply to a real life situation.

How do we draw out conceptual understandings from our students?

John Mason (1996) encapsulates the essence of math teaching in the following quote: a lesson without the opportunity for learners to generalise is not a mathematics lesson”.

In a concept-based model of curriculum and instruction we do not tell students the generalizations at the beginning of a unit of work or at the beginning of a lesson. We guide students to communicate generalizations through a variety of strategies.

The following table illustrates some examples of how we can draw conceptual understandings from our students.

Strategy Example

Give the concepts and ask students to connect these concepts in a statement that reflects the purpose of the unit of study.

Write the following concepts in a related sentence. This statement should demonstrate your understanding of how these concepts are connected:zeros of a function, roots of an equation, x intercepts, quadratic formulaor

parallel lines, gradient, transversal, angles

At the end of a lesson or unit of work ask students to complete the sentence, “I understand that……..”

I understand that solving a quadratic equation by using the quadratic formula graphically displays the x intercepts of a quadratic function.

Give students a choice of a few statements and ask students to choose one that explains the big idea of the lesson or unit and to justify why.

The quadratic formula isQuadratic EquationSolving a quadratic equation by using the quadratic formula graphically displays the x intercepts of a quadratic function.

The quadratic formula tells us what x is equal to and allows us to solve quadratic equations.

After a lesson or unit of study ask students to create a concept map, which includes the concepts they have learned and how they are connected with statements explaining this.

With the word quadratics at the centre, create a concept map using the ideas you have learned and include statements to explain how they are connected.

Give students a close activity where they have to fill in the blanks, and provide them with a list of hints.Hint Jar:

  • quadratic equation
  • quadratic formula
  • quadratic Function
  • x intercepts
Solving a ___________  ________by using the _________ ________ graphically displays the __________ of a ______________.
Ask students to devise a headline which summarizes the key ideas (concepts) of a unit. This can be used as a diagnostic or formative assessment.

Wathall, J. T. H. (2016) Concept-based mathematics: Teaching for deep understanding in secondary classrooms. Thousand Oaks, CA: Corwin.

Crafting generalizations is an important part of unit planning and if teachers know what their end goals are for their students then the rest of the unit planning process is enhanced.