PurposesEffective teaching; Lesson activity; Student elaboration

DescriptionQuestions are an important part of learning and teaching; a form of communication to gain insight into students’ conceptual understanding (Roth, 1996). According to Wolff-Michael Roth (1996), “teacher questions are frequent, pervasive and [an] universal phenomena.” But although a classroom norm, its use to promote conceptual understanding amongst students is scarce. The misconception that frequent questioning brings about conceptual understanding is prevalent amongst both pre-service and in-service teacher. These frequents questions only require students to memorize basic information, which is then recited. Based on the premise ‘quality not quantity,’ teachers need to ask “good” questions and allow students ample time to formulate a justifiable response. But what defines a “good” question?

Questions can be categorized according to different perspectives, the most common being Bloom’s Taxonomy. Bloom (1956) created a hierarchy of educational domain to which teachers can set educational objectives for students: cognitive, affective, psychomotor. The cognitive domain encompasses recollection, understanding, application, analysis, evaluation and creation (Airasian et al., 2001) . In order for students to successfully progress through each level they must have attained the prerequisite knowledge and skill in the preceding levels. Roth endorses that “good” questions: (1) provoke thought, (2) are based in students’ experiences, and (3) call for critical thinking (King, 1994). With the development of critical thinking, students are able to continuously analyze, evaluate and construct their thinking. Roth (1996) further believes, in order to bring about conceptual understanding, teachers must use questions to elicit justification, elaboration on previous answers and ideas, and predict contradictions to students’ intuitive ideas. Unfortunately, most teachers “request explicit, factual information” that does not require students to extend their thinking beyond memorization (Roth, 1996).

In the classroom sessions from which this analytic was built, Researcher Robert Davis of Rutgers University facilitated several research sessions in a sixth grade classroom at Harding Elementary School (K-8) in Kenilworth, New Jersey. The sessions are part of a longitudinal study of children’s mathematical thinking as they work on challenging tasks in environments designed to promote the building of powerful mathematical ideas or concepts through thoughtful doing of mathematics. The Rutgers-Kenilworth Longitudinal Study is from the Robert B. Davis Institute for Learning at Rutgers University in New Brunswick, New Jersey. The goal of this analytic is to provide pre-service and in-service educators with an opportunity to observe effective questioning techniques that promote conceptual understanding, in addition to the implementation select Common Core State Standards for Mathematical Practice (2010).

CCSS.MATH.PRACTICE.MP3. Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP7. Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8. Look for and express regularity in repeated reasoning.

In the beginning of the session, Researcher Robert Davis introduces the concept of variables via “boxes” and “triangles,” and guides students through solving quadratic equations. Based on question sequencing, students transition from “Guess-and-Check” to a refined methodology that takes into account the relationship between the constant and its factors. Given a task, students often implement random methods (i.e., trial and error, guess and check), however, in order to provide adequate justification for their solution, a transition is made to the use of a more systematic methodology (Maher & Martino, 1999).

Good questions must be accompanied with good questioning behaviors within a collaborative mathematical community, in order to promote conceptual understanding. In Event 2, Davis fosters an opportunity to build such an environment. Establishing these socio-mathematical norm in the beginning of class enable thorough discussion of expectations for class participation,

Davis also introduces the concept of the “secret” which, in its elementary stage, symbolizes the solution. In the selected events, notice how Davis progressively develops this concept to symbolize the relationship between the input (“box”) and the output (“triangle”).

During the sessions, through his sequencing of questions and by asking students to share “secrets,” Davis continually assesses students’ thinking and encourages them to construct their own procedures. His interventions were frequently in the form of encouragement for students to pursue ideas and think deeply, as teachers should guide rather than give instruction. Davis often observes without interruption and listens with interest as students share their idea. He also encourages students to reorganize their work and build connections, which helps them to bring about more sophisticated justifications and appropriate generalizations. Furthermore, Davis delays closure on solutions when students require more time to think and convince themselves of the reasonableness of their ideas (Maher & Martino, 1999). Events selected illustrate the characteristics of learning environments that encourage mathematical reasoning as described by Maher & Martino (1999). Such an environment:

1) Allows students time and provides opportunities for explorations and reinvention.

2) Embraces the ideal that students must express their current thinking.

3) Revisits previous or isomorphic tasks.

---

The following 6th Grade Common Core State Standards for Mathematical Content can be observed in this analytics (CCSSI, 2010, p.43-44).

Domain: Expressions and Equations (6.EE)

Cluster: Reason about and solve one-variable equations and inequalities.

CSS.MATH.CONTENT.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.MATH.CONTENT.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions.

CCSS.MATH.CONTENT.6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

Cluster: Represent and analyze quantitative relationships between dependent and independent variables.

CCSS.MATH.CONTENT.6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

---

Students in this analytics are shown working on the following problems:

(B) - Box | (T) - Triangle

Problem 2

B | T

0 | 5

1 | 7

2 | 9

3 | 11

4 | 13

Problem 3

B | T

0 | 1

1 | 4

2 | 7

3 | 10

Problem 6

B | T

0 | 1

1 | 2

2 | 5

3 | 10

4 | 17

5 | 26

Problem 9

B | T

0 | 0

1 | 1/2

2 | 2

3 | 4 1/2

4 | 8

5 | 12 1/2

6 | 18

---

References:

Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pintrich, P. R., Raths, J., & Wittrock, M. C. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives. Anderson LW and Krathwohl DR. New York: Addison Wesley Longmann.

Bloom, B. S.; Engelhart, M. D.; Furst, E. J.; Hill, W. H.; Krathwohl, D. R. (1956). Taxonomy of educational objectives: The classification of educational goals. Handbook I: Cognitive domain. New York: David McKay Company.

Common Core State Standards Initiative. (2010). Common Core state standards for mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

King, A. (1994). Guiding knowledge construction in the classroom: Effects of teaching children how to question and how to explain. American Educational Research Journal, 31 (2), 338 - 368.

Maher, C. A. (2010). The longitudinal study. In Combinatorics and Reasoning (p. 3-8). Springer Netherlands.

Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. The Journal of Mathematical Behavior, 18(1), 53-78.

Robert B. Davis Institute for Learning. (2014). Kenilworth Longitudinal Study. Retrieved from: http://www.rbdil.org/projects.html#kwls

Roth, W. M. (1996). Teacher questioning in an open‐inquiry learning environment: Interactions of context, content, and student responses. Journal of Research in Science Teaching, 33(7), 709-736.

Questions can be categorized according to different perspectives, the most common being Bloom’s Taxonomy. Bloom (1956) created a hierarchy of educational domain to which teachers can set educational objectives for students: cognitive, affective, psychomotor. The cognitive domain encompasses recollection, understanding, application, analysis, evaluation and creation (Airasian et al., 2001) . In order for students to successfully progress through each level they must have attained the prerequisite knowledge and skill in the preceding levels. Roth endorses that “good” questions: (1) provoke thought, (2) are based in students’ experiences, and (3) call for critical thinking (King, 1994). With the development of critical thinking, students are able to continuously analyze, evaluate and construct their thinking. Roth (1996) further believes, in order to bring about conceptual understanding, teachers must use questions to elicit justification, elaboration on previous answers and ideas, and predict contradictions to students’ intuitive ideas. Unfortunately, most teachers “request explicit, factual information” that does not require students to extend their thinking beyond memorization (Roth, 1996).

In the classroom sessions from which this analytic was built, Researcher Robert Davis of Rutgers University facilitated several research sessions in a sixth grade classroom at Harding Elementary School (K-8) in Kenilworth, New Jersey. The sessions are part of a longitudinal study of children’s mathematical thinking as they work on challenging tasks in environments designed to promote the building of powerful mathematical ideas or concepts through thoughtful doing of mathematics. The Rutgers-Kenilworth Longitudinal Study is from the Robert B. Davis Institute for Learning at Rutgers University in New Brunswick, New Jersey. The goal of this analytic is to provide pre-service and in-service educators with an opportunity to observe effective questioning techniques that promote conceptual understanding, in addition to the implementation select Common Core State Standards for Mathematical Practice (2010).

CCSS.MATH.PRACTICE.MP3. Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP7. Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8. Look for and express regularity in repeated reasoning.

In the beginning of the session, Researcher Robert Davis introduces the concept of variables via “boxes” and “triangles,” and guides students through solving quadratic equations. Based on question sequencing, students transition from “Guess-and-Check” to a refined methodology that takes into account the relationship between the constant and its factors. Given a task, students often implement random methods (i.e., trial and error, guess and check), however, in order to provide adequate justification for their solution, a transition is made to the use of a more systematic methodology (Maher & Martino, 1999).

Good questions must be accompanied with good questioning behaviors within a collaborative mathematical community, in order to promote conceptual understanding. In Event 2, Davis fosters an opportunity to build such an environment. Establishing these socio-mathematical norm in the beginning of class enable thorough discussion of expectations for class participation,

Davis also introduces the concept of the “secret” which, in its elementary stage, symbolizes the solution. In the selected events, notice how Davis progressively develops this concept to symbolize the relationship between the input (“box”) and the output (“triangle”).

During the sessions, through his sequencing of questions and by asking students to share “secrets,” Davis continually assesses students’ thinking and encourages them to construct their own procedures. His interventions were frequently in the form of encouragement for students to pursue ideas and think deeply, as teachers should guide rather than give instruction. Davis often observes without interruption and listens with interest as students share their idea. He also encourages students to reorganize their work and build connections, which helps them to bring about more sophisticated justifications and appropriate generalizations. Furthermore, Davis delays closure on solutions when students require more time to think and convince themselves of the reasonableness of their ideas (Maher & Martino, 1999). Events selected illustrate the characteristics of learning environments that encourage mathematical reasoning as described by Maher & Martino (1999). Such an environment:

1) Allows students time and provides opportunities for explorations and reinvention.

2) Embraces the ideal that students must express their current thinking.

3) Revisits previous or isomorphic tasks.

---

The following 6th Grade Common Core State Standards for Mathematical Content can be observed in this analytics (CCSSI, 2010, p.43-44).

Domain: Expressions and Equations (6.EE)

Cluster: Reason about and solve one-variable equations and inequalities.

CSS.MATH.CONTENT.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.MATH.CONTENT.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions.

CCSS.MATH.CONTENT.6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

Cluster: Represent and analyze quantitative relationships between dependent and independent variables.

CCSS.MATH.CONTENT.6.EE.C.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

---

Students in this analytics are shown working on the following problems:

(B) - Box | (T) - Triangle

Problem 2

B | T

0 | 5

1 | 7

2 | 9

3 | 11

4 | 13

Problem 3

B | T

0 | 1

1 | 4

2 | 7

3 | 10

Problem 6

B | T

0 | 1

1 | 2

2 | 5

3 | 10

4 | 17

5 | 26

Problem 9

B | T

0 | 0

1 | 1/2

2 | 2

3 | 4 1/2

4 | 8

5 | 12 1/2

6 | 18

---

References:

Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pintrich, P. R., Raths, J., & Wittrock, M. C. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives. Anderson LW and Krathwohl DR. New York: Addison Wesley Longmann.

Bloom, B. S.; Engelhart, M. D.; Furst, E. J.; Hill, W. H.; Krathwohl, D. R. (1956). Taxonomy of educational objectives: The classification of educational goals. Handbook I: Cognitive domain. New York: David McKay Company.

Common Core State Standards Initiative. (2010). Common Core state standards for mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

King, A. (1994). Guiding knowledge construction in the classroom: Effects of teaching children how to question and how to explain. American Educational Research Journal, 31 (2), 338 - 368.

Maher, C. A. (2010). The longitudinal study. In Combinatorics and Reasoning (p. 3-8). Springer Netherlands.

Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. The Journal of Mathematical Behavior, 18(1), 53-78.

Robert B. Davis Institute for Learning. (2014). Kenilworth Longitudinal Study. Retrieved from: http://www.rbdil.org/projects.html#kwls

Roth, W. M. (1996). Teacher questioning in an open‐inquiry learning environment: Interactions of context, content, and student responses. Journal of Research in Science Teaching, 33(7), 709-736.

Created on2014-04-28T14:14:12-0400

Published on2015-07-24T09:16:27-0400

Persistent URLhttp://dx.doi.org/doi:10.7282/T34Q7WS9