Good Questions for Math Teaching: Creating Effective Questions for Math Education, Lecture notes of Mathematics

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Good uestions for Math Teaching

Why Ask Them and What to Ask K–

Peter Sullivan

Pat Lilburn

Math Solutions Publications Sausalito CA

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ontents

vii Acknowledgments

1 PART ONE: THE IMPORTANCE OF QUESTIONING

3 What Are Good Questions? 7 How to Create Good Questions 11 Using Good Questions in Your Classroom

17 PART TWO: GOOD QUESTIONS TO USE IN MATH LESSONS

19 Number 20 Money 25 Fractions 29 Decimals 33 Place Value 36 Counting and Ordering 40 Operations

47 Measurement 48 Weight 51 Volume and Capacity 56 Area 61 Time 66 Length and Perimeter

73 Space 74 Location and Position 77 Two-Dimensional Shapes 84 Three-Dimensional Shapes

91 Chance and Data 92 Chance 97 Data

C

cknowledgments

The idea of open-ended or good questions developed over several years during ongoing discussions between Peter Sullivan and David Clarke. Peter and David conducted a number of research studies and classroom trials of open-ended questions. Many of David’s initial ideas are used in various places throughout this book. His creativity, energy, and interest in exploring good questions contributed significantly to the idea of using open-ended activ- ities in the teaching of mathematics, and for this we thank him. The idea went through a number of phases before it reached its final form. Pam Rawson’s contribution to the early planning stages, which ultimately led to the development of this resource, is greatly appreciated. Finally, we thank Sheryl and Mike without whose continued support there would be no book.

PETER SULLIVAN PAT L ILBURN

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In this book we describe the features of good questions, show how to create good questions, give some practical ideas for using them in your class- room, and provide many good questions that you can use in your mathe- matics program.

2i   Good Questions for Math Teaching

What Are Good Questions?

Let us have a closer look at what makes a good question. There are three main features of good questions.  They require more than remembering a fact or reproducing a skill.  Students can learn by answering the questions, and the teacher learns about each student from the attempt.  There may be several acceptable answers. This section explains these features in more detail.

More Than Remembering

A particular grade 6 student, Jane, had just finished a unit on measurement where she had been asked to calculate area and perimeter from diagrams of rec- tangles with the dimensions marked. She was able to complete these correctly, and the teacher assumed from this that Jane understood the concepts of area and perimeter. However, when she was asked the following good question she claimed that she could not do it because there was not enough information given. I want to make a garden in the shape of a rectangle. I have 30 meters of fence for my garden. What might be the area of the garden? To find an answer to this Jane needed to think about the constraints that a perimeter of 30 meters places on the lengths of the sides of the rectangle, as well as thinking about the area. She needed to use higher order reasoning skills since she had to consider the relationship of area and perimeter to find possible whole number answers that could range from 14 x 1 (14m^2 ) to 7 x 8 (56m 2 ). This certainly required her to do more than remember a fact or reproduce a skill. It required comprehension of the task, application of the concepts and appro-

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to suggest what they need to think about when measuring length. Their list included the need to:

 agree on levels of accuracy  agree on where to start and finish, and the importance of starting at the zero on the yardstick  avoid overlap at the ends of the yardsticks  avoid spaces between the yardsticks  measure the shortest distance in a straight line.

By answering the question the students established for themselves these essential aspects of measurement, and thus learned by doing the task. As we have discussed, the way students respond to good questions can also show the teacher if they understand the concept and can give a clear indication of where further work is needed. If Jane’s teacher had not presented her with the good question she would have thought Jane totally understood the concepts of area and perimeter. In the above example, the teacher could see that the children did understand how to use an instrument to measure accurately. Thus we can see that good questions are useful as assessment tools, too.

Several Acceptable Answers

Many of the questions teachers ask, especially during mathematics lessons, have only one correct answer. Such questions are perfectly acceptable, but there are many other questions that have more than one possible answer and teachers should make a point of asking these, too. Each of the good questions that we have already looked at has several possible answers. Because of this, these ques- tions foster higher level thinking because they encourage students to develop their problem-solving expertise at the same time as they are acquiring mathe- matical skills. There are different levels of sophistication at which individual students might respond. It is characteristic of such good questions that each student can make a valid response that reflects the extent of their understanding. Since cor- rect answers can be given at a number of levels, such tasks are particularly appro- priate for mixed ability classes. Students who respond quickly at a superficial level can be asked to look for alternative or more general solutions. Other stu- dents will recognize these alternatives and search for a general solution. If we think back to the earlier question on the area of the garden, there is a range of acceptable whole number answers (14 x 1, 13 x 2, 12 x 3... 8 x 7). Students could be asked to find the largest or smallest garden possible. They

5i   What Are Good Questions?

could be asked to describe all possible rectangles. Other students will be inter- ested in exploring answers other than those that involve only whole numbers, for example, 12.5m x 2.5m. It is the openness of the task that provides this rich- ness. The existence of several acceptable answers stimulates the higher level thinking and the problem solving.

In this section, we have looked more closely at the three features that categorize good questions. We have seen that the quality of learning is related both to the tasks given to students and to the quality of questions the teacher asks. Students can learn mathematics better if they work on questions or tasks that require more than recall of information, and from which they can learn by the act of answering the question, and that allow for a range of possible answers. Good questions possess these features and therefore should be regarded as an important teaching tool for teachers to develop. The next section shows two ways to construct your own good questions.

6i   Good Questions for Math Teaching

Step 2: The closed question might be The children in the Smith family are aged 3, 8, 9, 10, and 15. What is their average age? The answer is

Step 3: The good question could be There are five children in a family. Their average age is 9. How old might the children be?

Some more examples of how this works are shown in the following table.

Method : Adapting a Standard Question

This is also a three-step process. Step 1: Identify a topic. Step 2: Think of a standard question. Step 3: Adapt it to make a good question.

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STEP STEP  STEP  Identify a topic Think of an answer Make up a question that includes the answer rounding 11.7 My coach said that I ran 100 yards in about 12 seconds. What might the numbers on the stopwatch have been? counting 4 chairs I counted something in our room. There were exactly 4. What might I have counted? area 6cm^2 How many triangles can you draw each with an area of 6cm^2? fractions 3  Two numbers are multiplied to give 3 . What might the numbers be? money 35 cents I bought some things at a supermarket and got 35 cents change. What did I buy and how much did each item cost? graphing x What could this be the graph of? x x x x x x x x x x x x x x x x 1 2 3 4 5

For example: Step 1: The topic for tomorrow is measuring length using nonstandard units. Step 2: A typical exercise might be What is the length of your table meas- ured in handspans? Step 3: The good question could be Can you find an object that is three handspans long? Some more examples of how this works are shown in the following table.

The more experience you have with good questions the more you will want to use them, and the easier it will become for you to make up your own. Refer to either or both of these methods until you feel confident.

9i   How to Create Good Questions

STEP STEP  STEP  Identify a topic Think of an standard Adapt it to make a good question question

space What is a square? How many things can you write about this square?

addition 337 + 456 = On a train trip I was working out some dis- tances. I spilt some soft drink on my paper and some numbers disappeared. My paper looked like 3? 7 +?? 6 7 9? What might the missing numbers be? subtraction 731 – 256 = Arrange the digits so that the difference is between 100 and 200. time What is the time What is your favorite time of day? shown on this clock? Show it on a clock.

Using Good Questions in Your Classroom

Today’s mathematics classrooms should be dynamic places where children are involved and engaged in their own learning. This can be achieved through activ- ities that promote higher level thinking, cooperative problem solving, and com- munication. We have seen that good questions support these activities and are readily available for teachers to use. The first part of this section describes generally how to use a good question as the basis of a mathematics lesson. It sets out the impor- tant steps of the lesson, explains the roles of the teacher and students, and advises how to overcome problems that could arise at each stage. The second part of this section takes you through each of the steps with a specific good ques- tion. Before the start of a lesson it is necessary to choose or create a good ques- tion. This should be aimed at the appropriate level for the children in your class. At first you might find the question you choose is too easy or too difficult, but keep practicing because you will soon get the hang of it. Once you have chosen the question then the following steps should help you to use it with your class.

Step : Pose the Good Question

It is a good idea to have the question written on the blackboard and as you ask

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the question refer to the words on the board. It is very important to make sure that all children know what the question is; do not assume they know it because it is on the board. You could even ask some students to repeat the question in their own words. Allow some time for children to ask you about the meaning of the task. Explain the task to them if necessary but do not give any directions or sugges- tions on how to do it. This is for the children to work out for themselves.

Step : Students Work on the Good Question

When first using good questions in your classroom it is better to let the children work in pairs or small groups. This allows them to communicate their ideas to oth- ers. This communication is an important part of learning. Working together can also assist those children who may have difficulty starting. If these children have to wait for the teacher then organizational and attitudinal problems can arise. If, once children start working, there are too many who cannot make progress without teacher assistance then it might be necessary to stop and have a whole class discussion to overcome the general concerns. If the con- cerns of each group, or individuals within each group, are all different then this is a sign that the question you have posed is too difficult for the class. If this happens either make the question easier or suggest that the students represent the problem in some way, such as by using materials or drawing a diagram. A variety of concrete materials should always be available for chil- dren to select from. You could also decide to abandon the question alto- gether as unsuitable at this stage. If this happens do not worry, as it takes time and practice to choose appropriate good questions. However, you will find good questions to be worth the effort and perseverance. Ideally, you should plan in advance how to help children who may not be able to start on the question. Once the groups are working, your task is to monitor their progress. If a group stops after giving one response, ask them to look for other possible answers. If they have found all possible answers ask them to describe all their answers. In this way they can experience the meaning of a general solution. You could also ask a related question to extend them. For example, a related ques- tion for the task The stopwatch shows tenths of seconds. My coach said that I ran 100 yards in about 12 seconds. What might the numbers on the stopwatch have been? could be What if the stopwatch showed hundredths? It is not vital that you wait until all groups have finished the task before initiating a discussion. They will all have answered the question to a degree. It

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