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Open Questions for Every Student

Marian Small

In this issue’s feature article, “Open Questions for Every Student,” first published more than two years ago, you’ll hear more from me about open questions, particularly how they can help you in setting up a classroom that is attentive to student needs and ultimately more equitable for all students.

One issue that is front and centre in Canada right now is the question of equity. Equitable instruction ensures that every student is presented with opportunities to build success in an appropriate learning environment.

It has been my belief for a long time that one of the best ways to provide equitable instruction in mathematics is to use open questions. When you use open questions, you ensure that the tasks presented are appropriate for and accessible to each and every learner.

 

    What Are Open Questions?

    Open questions are questions that are not specific enough to have only one answer or response.

    Contrast, for example, this question:

      •  What is 6 × 4?

      with these open questions:

        • You multiply two numbers that are two apart. What might the answer be?
          [E.g., 4 × 6 is 24; 1 × 3 is 3; 10 × 8 is 80]
        • You multiply two even numbers. What might the answer be?
          [E.g., 4 × 6 = 24; 2 × 2 = 4; 4 × 2 = 8]
        • You multiply two numbers, and the answer is in the 20s. What might the numbers be?
          [E.g., 3 × 7 is 21; 4 × 6 is 24; 2 × 10 is 20]
        • You multiply two numbers that are less than 10. What might the answer be?
          [E.g., 1 × 1 is 1; 4 × 6 is 24; 5 × 5 is 25]

        All five questions address multiplication, but the last four are more likely to lead to conversations in which many students can share a variety of perspectives and ideas. In comparison, if you were to pose the first question, it would not likely achieve the same result.

        How Do Open Questions Promote Equity?

        Consider an open question such as “You multiply two even numbers. What might the answer be?”

        A student who is insecure might use a very simple pair of numbers, such as 2 and 2, but other students might wonder what would happen if they used bigger numbers. Some students might even wonder what all the possible answers or non-answers are. All students have the opportunity for success, and, generally, students will benefit from other students’ thinking as well as their own. Open questions create rich discussions that every student can benefit from.

        What Makes for a Good Open Question?

        An open question is most valuable if

          • It is appropriate for the curriculum at the grade level you are teaching.
          • It is suitable for a broad range of students.
          • It stimulates new thinking and does not just test whether students already know something.
          • It has the potential to lead to rich discussion and follow-up work.

          Let’s look at some examples of open questions at various grade levels.

          The Open Question

          Use Cuisenaire rods.
          Make a line with a short rod and a long rod.
          Make a line with two or three other rods that is exactly the same length as the first line.
          Write addition equations to show what happened.

          Logistics
          • You might provide a link to online Cuisenaire rods, or you might provide actual rods. An alternative is to use linking-cube trains that have lengths from 1 to 10, ideally with each length a different colour.
          • You might show students how to make a line, but avoid showing them the answer, which could result in some students just copying your answer without thinking about the activity. You might also want to demonstrate how to write the total length as an addition equation (e.g., 9 + 3 = 12 for the first line in the illustration).
          Success Criteria

          You might provide success criteria. For example, students must do the following:

          1. Create at least five examples.
          2. Use each rod length from 1 to 10 as part of at least one example.
          3. Use two rods in the second line some of the time, and use three rods in the second line other times.
          4. Say a few things they noticed about whether the lines ended up including long, short, or medium rod lengths.
          Relevance to the Curriculum

          One objective of this question is to get students to practise addition facts and concepts and notice addition principles. This objective is relevant to the curriculum in Kindergarten and Grade 1 and possibly Grade 2 since it solidifies the learning of addition and subtraction facts. For example, the two lines below show that adding 9 to a number is like adding 10 and 1 less than the other number.

          blue 9

          green 3

          orange 10

          red 2

          Another goal is for students to practise writing mathematical equations to represent addition expressions (e.g., 9 + 3 = 10 + 2), which helps students see that equations are about balance, not about simply getting answers.

          Suitability for a Broad Range of Students to Promote Equity

          In this question, students are working directly with the materials and have a choice of different rods to work with, so the task should be accessible to almost every student. It also allows for the potential to lead to mathematical generalizations such as the associative property of addition; for example, (9 + 1) + 2 = 9 + (1 + 2).

          How Is New Thinking Stimulated?

          Rather than just asking students if they know whether particular addition combinations are equivalent, this task helps students actually learn those combinations through using the rods.

          Discussion Time

          After students have worked on the task, you might start the conversation with questions such as these:

          • Did anyone use a blue rod in one of their lines? Tell us one of the colours in your other line, and we will try to guess your equation.
          • Did anyone use two long rods in one line but only short rods in the other? [Note: This should not be possible.]
          • Did anyone use two medium-sized rods to make the same length as a long rod and a short rod together?
          • Did anyone have an equation that showed doubles?
          Potential Follow-Up

          Notice that the responses to these questions could easily lead to further activities. Here are some you might ask your students to try:

          • Use your rods to show that every even number can be shown as a line of two rods that are the same colour.
          • Use your rods to show that every odd number can be shown as a line of three rods, including two of the same colour.
          • Use your rods to show that to add 9 to a number, you can always, instead, add 10 to a number that is 1 less than that number.

          The Open Question

          A lot of coins are worth about $8 altogether.
          What might the coins be?
          How did you figure it out?
          How do you know you are correct?

          Logistics
          • You might first remind students of what coins they can use.

          Success Criteria

          You might provide success criteria. For example, students must do the following:

          1. Create at least four or five possibilities.
          2. Create a short video to explain how they came up with their ideas.
          3. Demonstrate how they would count the coins to show they are correct.
          Relevance to the Curriculum

          One objective is for students to practise determining the value of a collection of coins by using knowledge of equivalences (e.g., 4 quarters is equal to 1 loonie), knowledge of how to count efficiently, and knowledge of addition. This is relevant to the curriculum at different grade levels in different jurisdictions.

          There are many correct answers to this question; for example:
          78 dimes
          OR
          7 loonies and 9 dimes
          OR
          61 nickels and 5 loonies
          OR
          9 nickels, 10 quarters, and 5 loonies

          Suitability for a Broad Range of Students to Promote Equity

          Students can stick with dimes, loonies, and toonies to make calculations simpler, but there are so many combinations that there is the potential for students who want more challenge to consider a much broader range of possibilities. The use of the phrase “a lot of coins” allows for even more access for all students.

          How Is New Thinking Stimulated?

          This is not a quick task—as “How many quarters make a dollar?” would be. It is a task that requires students to put together information they might already know.

          Discussion Time

          If students share their work with you individually, you might choose a few of their responses to talk about as a whole class. For example, you could ask the following:

          • Why couldn’t you use all toonies? What is the greatest number of toonies you might use? Why?
          • Could you use mostly quarters? Did anyone?
          • Could you use only loonies and quarters?
          • Would you use a lot of pennies? Why or why not?
          • Did anyone get exactly $8? Is that possible?
          Potential Follow-Up

          You could extend this activity by posing tasks such as these:

          • How could you use just a few coins to create an amount worth $5?
          • How could you use a lot of bills to create an amount worth about $100?
          • If you use 20 coins to create an amount worth $8, what could 21 coins be worth?

          The Open Question

          You write down a three-digit, four-digit, or five-digit number.
          You switch some of the digits around, and your new number is greater than your original number by one of these values:
          19 800 or
          1980 or
          198

          What could your original number have been?
          What is your new, switched-around number?
          How do you know?

          Logistics
          • You might have a brief meeting with students to explain what you mean by showing them that if you switch the digits of 42 153 to 41 235, the value of the new number has decreased (e.g., 42 153 – 41 235 = 918). Make sure that students realize that the outcome of this example is different since the value decreases instead of increases as it is meant to do in the original question.
          Success Criteria

          You might provide success criteria. For example, students must do the following:

          1. Write down both the original number and the new, switched-around number.
          2. Prove that the change is an increase of either 19 800, 1980, or 198.
          3. Talk about their strategy in finding an answer.
          4. Tell whether they think there are many more possible answers and why or why not.
          Relevance to the Curriculum

          One objective of this question is to prompt students to consider place value. Notice that 19 800 is 20 000 – 200, 1980 is 2000 – 20, and 198 is 200 – 2. In each case, one digit of the number is increased by 2 (either the ten thousands digit, the thousands digit, or the hundreds digits) and one is decreased by 2 (either the hundreds digit, the tens digit, or the ones digit). This problem very much supports work in place value at higher elementary grades.

          Suitability for a Broad Range of Students to Promote Equity

          This question was designed to be suitable for many students by allowing for a choice of three-digit, four-digit, or five-digit numbers, but the same idea is explored in each case. That will mean the discussion will benefit students who might have shied away from five-digit numbers.

          How Is New Thinking Stimulated?

          Rather than just answering quick questions such as “What is the digit 2 worth in 32 145?”, students must use ideas about how the placement of a digit in a number affects its value in a problem-solving situation.

          Discussion Time

          After students have worked on the task, you could ask questions such as these:

          • Were there any digits in your first number that were not in your second one? [Note: the answer should be no; you might ask this question to confirm the constraints of the problem.]
          • Did any digits stay in exactly the same spots? Which ones?
          • Did any digits increase? Which ones? By how much?
          • Did any digits decrease? Which ones? By how much?
          • Why does what happened to your new number make sense?
          • How do you know that there could be more than one answer?
          Potential Follow-Up

          Notice that the responses to the above questions could easily lead to further activities. For example, you could have students try the following:

          • Suppose you switch the digits of a four-digit or five-digit number. Could the number
            – increase by 20 200?
            – decrease by 1700?
            – increase by 294?
            – decrease by 600? How?
          • How else could the values increase or decrease?

          The Open Question

          On a number line, there are four negative integers: A, B, C, and D.

          A and B are twice as far apart as B and C.
          B and C are three times as far apart as C and D.

          What could the integers be?

          Logistics
          • You might have a brief discussion with students to show them what you mean by saying that one pair of integers is twice as far apart as another pair. You could ask students to help you demonstrate this same concept on the positive side of a number line (e.g., 2 and 4 versus 100 and 104).
          Success Criteria

          You might provide success criteria. For example, students must do the following:

          1. Create at least five examples.
          2. Use only negative integers.
          3. Be ready to discuss their thinking.
          Relevance to the Curriculum

          One objective is to have students practise placing negative integers on a number line while also recognizing the spacing between integers. This kind of practice is appropriate in grades in which students work with integers.

          Suitability for a Broad Range of Students to Promote Equity

          This question is suitable for many students because it uses simple multipliers such as 2 and 3 (twice as far apart and three times as far apart), and it allows students to choose to work with integers close to or far from 0.

          How Is New Thinking Stimulated?

          Again, this task does not simply ask a question such as “What is an integer 20 less than –2?” but asks students to solve a problem that incorporates integer addition or subtraction.

          Discussion Time

          After students have worked on the task, you could ask questions such as these:

          • Did anyone have C and D one unit apart? Where were all of your integers? [Note: Some examples of sets of integers that meet these criteria are –1, –7, –10, –11; –10, –16, –19, –20; and –30, –24, –21, –20.]
          • Did anyone have D and C farther apart than A and B? [Note: The answer should be no.]
          • Could A have been –100?
          • Could A have been –1?
          Potential Follow-Up

          Notice that the responses to these questions could easily lead to further activities, such as the following:

          • Argue whether A must always be on one end of the number line. Why or why not?
          • How would the results and the order of the integers change if these were the conditions instead:
            A and C are twice as far apart as B and D.
            C and D are three times as far apart as B and D.
          • Change the conditions another way, and see how it affects the resulting order of the integers.