Students who have difficulty with mathematics need many opportunities to respond to effective questions, explain their thinking, and receive feedback that allows them to improve their learning. To increase generalization of skills and flexibility in thinking mathematically, instructors need to ask questions that increase student engagement, that provide feedback, and that are linked to algebraic or higher level thinking and understanding (Cai & Knuth, 2005; Witzel, Mercer, & Miller, 2003). Specifically, beginning algebraic thinking, such as analyzing relationships, generalizing models, predicting, justifying, or noticing structure, can lead to greater gains in mathematics understanding in later years (Kieran, 2004).
Following are guidelines for asking questions that will move student learning forward, increase student engagement, and offer immediate feedback.
Questioning: The three main types of questions that should be used in mathematics are reversibility, flexibility, and generalization (Dougherty, Bryant, Bryant, Darrough, & Pfannenstiel, 2015).
- Reversibility questions are those that change the direction of student thinking: for example, giving the student the answer and asking him or her to identify the correct equation. This type of question allows for multiple answers and gives students the opportunity to think about algorithms in different ways. Reversibility questions should be presented after the student has demonstrated mastery of a particular procedure or algorithm.
- Flexibility questions support student understanding in finding multiple ways to solve a problem or in discerning relationships among problems. For example, the student might be asked to solve an addition problem using a specific strategy and then show or prove the answer using another method, such as a number line. Flexibility questions can be used during instruction to show relationships between similar problems or differences in models.
- Generalization questions are those that ask students to create statements about patterns. In the past, instructors would explain algorithms or rules, and they did not afford students the opportunity to develop explanations on their own. To increase conceptual understanding, guided questions about patterns allow rules or generalizations to be “discovered” by the student. For instance, students are presented with a list of numbers multiplied by two and then asked to describe any patterns they notice (e.g., one factor is two, product is an even number, etc.). The use of generalization questions allows students to develop a deeper understanding of mathematics and to generalize their thinking to similar problems.
Regardless of the type of questions asked, instructors should use the questioning strategy to assess student understanding and then use the information obtained from the questioning to evaluate and adjust their instruction as necessary.
Teachers should involve all students in questioning. This involvement can be accomplished in several ways:
First, teachers may invite all students to respond to questioning through unison choral response. Although this is an easy way to encourage students to respond, it is important to ensure that all students are responding to the questions at the same time.
Second, teachers can use equity sticks. Teachers write each student’s name on an ice pop stick and then draw a stick for each question they ask. The student whose name appears on the stick answers the question. All students have the same chance to be called upon.
Third, teachers may use response cards. Write “A,” “B,” and “C” on separate cards. The instructor asks a question and presents three answer choices. Students select their choice and hold up the response card indicating their answer.
Fourth, teachers may ask students to write their answers on whiteboards. Students hold up the answers so the instructor can check them for accuracy.
Fifth, teachers may invite students to create a model. Students then pair-share their creations to identify differences and similarities among the models and answers to the mathematical questions.
Teachers may need to individualize their questions for students to gain a better understanding of a particular student’s knowledge of the skill that is being taught.
Feedback: Providing students with both positive and corrective feedback is essential to their learning. It is important that students receive immediate feedback so that they do not continue to practice incorrectly. Students should also have an opportunity to practice/repeat the correct response after error correction has been provided (Archer & Hughes, 2011).
