Insights Into Algebra 1: Teaching for Learning
Direct and Inverse Variation Teaching Strategies: Questioning Techniques
Educational philosopher John Dewey said: “What’s in a question, you ask? Everything. It is a way of evoking stimulating response or stultifying inquiry. It is, in essence, the very core of teaching.”
Effective questions are an integral part of the successful mathematics classroom. Some research suggests that as much as 50 percent of classroom time is spent asking questions and eliciting responses. Instruction that includes questions during lessons is more effective in producing achievement gains than instruction carried out without putting questions to students.
In fact, the National Council of Teachers of Mathematics (NCTM) believes so strongly in effective questions as a way of promoting understanding that their Principles and Standards for School Mathematics (PSSM) often suggests which questions to ask. For a problem about cell phone rates, the PSSM recommends:
Before the students solve the problem, a teacher might ask them to use their table and graph to focus on important basic issues regarding the relationships they represent. By asking, ‘How much would each company charge for 25 minutes? For 100 minutes?’ the teacher could find out if students can interpret and extend the patterns. Since the table identifies only a small number of distinct points, a teacher could ask why it is legitimate to connect the points on the graph to make a line. Students might also be asked why one graph includes the origin but the other does not.
With so much time in the classroom dedicated to questioning, and with increased student learning as a potential outcome, every teacher should develop the ability to pose queries effectively.
Why Ask Questions?
The following is a partial list of the questions that teacher Peggy Lynn asked during her two lessons on direct and inverse variation. As you read each question below, think about Peggy’s purpose in asking it. Ask yourself, “Why did she ask that question?”
- How did you come up with your estimation?
- When you say “pattern,” what kind of pattern are you referring to?
- And what does that “+ 1” on the end mean?
- Any questions so far?
- Why did you do 100 � 100?
- You seem pretty certain of that. Why do you think it’s not [a direct proportion]?
- So how many gallons would there be in 920,000 barrels?
- Could a direct variation have a negative slope?
- If you have zero drops, how much area should you have?
- What just happened there, when you doubled your volume?
- What about if you made the area of the base get smaller and smaller, your diameter got smaller and smaller. What’s going to happen to the height of your water?
Questions in the math classroom serve a variety of purposes, from increasing student comprehension and clarifying student thinking, to aiding in social development. The following list gives many of the reasons teachers ask questions.
- To involve students in the lesson by letting them share ideas that provide clarification and a deeper analysis of problems.
Example: You seem pretty certain of that. Why do you think it’s not [a direct proportion]?
- To provide assessment of what students know to help guide instruction.
Example: Why did you do 100 � 100?
- To enhance retention of important information and to provide increased understanding of the major mathematical skills and concepts.
Example: If you have zero drops, how much area should you have?
- To aid in classroom management by redirecting discussions, making sure that students comprehend directions, and checking for understanding. (Many questions in this category are not prepared in advance – teachers ask them as the need arises.)
Example: Any questions so far?
What is your primary purpose for asking questions in class? Have a colleague attend one of your classes and generate a list of all the questions you ask during class. How do the questions you ask promote student learning of mathematics?
Using Questions Effectively
The Northwest Regional Educational Laboratory, in an article on effective questioning, claims that “[o]ral questions posed during classroom recitations are more effective in fostering learning than written questions.”
This is an astounding fact, and one that may be counterintuitive. An important corollary to this statement is that questions asked orally in class must be relevant and clear to students. With that in mind, the following are important guidelines for using questions effectively in the classroom:
- Ask challenging questions.
- Ask well-crafted, uncluttered, open-ended questions. Do not include lots of extraneous words, or information intended to lead students to the “correct response.” The lesson should give them the information they need, and questions should allow them to think about what they have learned.
- Ask questions that focus on the important parts of the lesson, not about unrelated topics.
- For lower cognitive questions, research has shown that teachers should allow three seconds of wait-time for students to decide on the answer. With higher cognitive questions, however, research has shown that the more wait-time given, the more thorough the responses will be.
- Ask only as many questions as necessary. During a difficult lesson, too many questions may actually have a deleterious effect.
- Avoid being vague or critical when responding to student answers. When offering praise, be sure that it is directed to the student’s response.
- Find a way to allow all students the opportunity to answer a question. Use a random technique for selecting students to avoid always calling on the same students.
Read Peggy Lynn’s reflections on her questioning strategies in the video lessons for Workshop 7:
Transcript from Peggy Lynn
I think it is critical for students to explain their thinking – for correct and incorrect responses. My job is to guide them through the process as they construct the mathematical concepts. Asking how and why, not just what, during discussions is helpful to get students to communicate their thoughts. When a student answers a question incorrectly, I may suggest the question that their response would have answered correctly, instead of just replying “wrong.” It takes time to get students to be comfortable with making educated guesses – to be risk takers. There is a lot of power in “figuring it out yourself.”
Based on these suggestions, list two or three techniques that you will try in your classes this week in regard to questioning.
Using Effective Questions
Lower cognitive questions usually elicit a short answer, perhaps a yes or no, or maybe just a short phrase from students. Lower cognitive questions give students a chance to demonstrate knowledge, yet they typically require little or no thought. Simple, factual questions usually have a right answer, and students either know the answer or they don’t.
On the other hand, higher cognitive questions allow for students to interpret and evaluate, and they require students to think, rather than just recall. They stimulate discussion, and they force students to a higher level of cognition in preparing a response. Higher cognitive questions often begin with the following phrases:
- What would happen if … ?
- What would have to happen for … ?
- What happens when … ?
It is often possible to turn one (or more) lower cognitive question into a higher-order question.
The lower cognitive question What is a direct variation? asks for recall of basic facts, namely the definition of a direct variation.
What are the important characteristics of a direct variation?
In addition to knowing the definition of a direct variation, a student must also understand the subtleties of a direct variation to answer this question. Better still, the following question requires students to evaluate and synthesize all of the information they’ve learned about direct variation and apply it to a new situation:
Can you give me an example of a direct variation?
You can combine the two lower cognitive questions What is a direct variation?and What is an inverse variation? into one higher cognitive question that could generate more discussion in the classroom:
What are the differences between a direct variation and an inverse variation?
Pick one of the following lower cognitive questions, and transform it into a higher cognitive question to generate a classroom discussion.
- What is an inverse variation?
- What is the constant of proportionality?
- What is the equation for a direct variation? For an inverse variation?
- What does m stand for?
- What does k represent?
Wait-Time and Pacing of a Lesson
An important component of using questions effectively is allowing enough time for students to formulate an adequate response. Often, teachers allow only one second after asking a question before calling on a student. Yet, research has shown that student achievement improves with more wait-time.
- For lower cognitive questions, a wait-time of three seconds is most positively related to achievement, with less success resulting from shorter or longer wait-times. A lower cognitive question might be How many inches are in a foot? or What is the value of ?
- There seems to be no wait-time threshold for higher cognitive questions; students become more engaged and perform better the longer the teacher is willing to wait. A higher cognitive question might be What is a real-world example of a situation that involves inverse variation? Source: Cotton, Kathleen. Classroom Questioning, Northwest Regional Educational Laboratory, 2001.
It is important to incorporate both higher and lower cognitive questions in lessons. Because the wait-time for lower cognitive questions is typically shorter, incorporating them helps vary the pace of a lesson. But higher cognitive questions require students to think, and they can often lead to insightful discussions.
For a lesson that you are teaching this week, make a list of five to 10 lower cognitive questions that you can ask to help with the pace of your lesson. In addition, craft two higher cognitive questions that require students to think deeply about the topic.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.