1. Good Questions: What are they & How to create them?
Dr. Sarah Ledford
Mathematics Educator

2. What are GOOD questions?
They require more than remembering a fact or reproducing a skill.
Students can learn by answering the questions.
Teachers learn about each student from their attempts (at the answer).
There may be several acceptable answers.
From Sullivan & Lilburn (pg 3).

3. What are GOOD questions? (updated answer)
They help students make sense of the mathematics.
They are open-ended, in answer or approach. There may be multiple answers or multiple approaches.
They empower students to unravel their misconceptions.
They not only require application of facts and procedures but encourage students to make connections and generalizations.

4. What are GOOD questions? (updated answer)
They are accessible to all students in their language and offer an entry point for all students.
Their answers lead students to wonder more about a topic and to perhaps construct new questions themselves as they investigate this newly found interest.
From Schuster and Anderson (pg 3).

5. Classroom Environment
Create a safe environment.
Students are taking a risk & need to not be discouraged.
Everyone must keep an open-mind.
Disagreements may occur.
Important to note that the disagreement is with the idea NOT the person who has the idea.
Treat others civilly & with respect.

6. Classroom Environment
Think about where a student will begin.
Give sufficient wait-time.
Discuss answers.
From Schuster and Anderson (pgs 9–11).

7. Teacher Questions Why do you think that?
How did you know to try that strategy?
Will this work with every number or in every situation?
When will this strategy not work? Can you give a counterexample?
Who has a different strategy?
How is your answer like or different from another student’s?
Can you repeat your classmate’s ideas in your own words?
Do you agree or disagree with your classmate’s idea? Why?
From Schuster and Anderson (pg 11).

8. Let’s make some good questions…
…by adapting a standard question.

9. Adapting a Standard Question
Step 1: Identify a topic.
Step 2: Think of a standard question.
Step 3: Adapt it to make a good question.
From Sullivan & Lilburn (pgs 8–9).

10. Adapting a Standard Question
Step 1: Identify a topic.
Measuring length using nonstandard units
Step 2: Think of a standard question.
What is the length of your table measured in handspans?
Step 3: Adapt it to make a good question.
Can you find an object that is three handspans long?
From Sullivan & Lilburn (pgs 8–9).

11. Adapting a Standard Question
Step 1: Identify a topic.
Telling time
Step 2: Think of a standard question.
What is the time shown on the clock?
Step 3: Adapt it to make a good question.
What is your favorite time of day? Show it on a clock.
From Sullivan & Lilburn (pgs 8–9).

12. Adapting a Standard Question
Step 1: Identify a topic.
Subtraction
Step 2: Think of a standard question.
731 – 256 = ?
Step 3: Adapt it to make a good question.
Rearrange the digits so that the difference is between 100 & 200.
From Sullivan & Lilburn (pgs 8–9).

13. Answer these questions like a student might…
…without just saying IDK, etc. ;)

14. Adding Integers Annie added three integers and got a sum of zero.
What might the integers have been?
From Schuster & Anderson (pg 26).

15. From Schuster & Anderson (pg 26).
Adding Integers
Hint: Think of a pair of opposites & write one as the sum of two integers.
For example,
= 0
= 0
From Schuster & Anderson (pg 26).

16. From Schuster & Anderson (pg 56).
Dividing Fractions
When dividing fractions, can your answer be greater than, less than, or between the two fractions you are dividing?
Explain why or why not for each situation & give examples to illustrate your position.
From Schuster & Anderson (pg 56).

17. From Schuster & Anderson (pg 56).
Dividing Fractions
Greater than:
¾ ÷ ¼ = 3
Less than:
¾ ÷ 2 = 3/8
In between:
¼ ÷ ¾ = 1/3
From Schuster & Anderson (pg 56).

18. Writing Equations
The students in Ms. Sikes’ math class were playing Guess My Rule. Ms. Sikes showed the students pairs of starting and final values and asked the students to use the information to determine the rule.
Ms. Sikes wrote the first pair in the extended T-chart as shown below.
What could the rule be? Do you know for sure?
Starting Value Using the Rule Final Value
From Schuster & Anderson (pg 105).

19. Writing Equations
Starting Value Using the Rule Final Value 4 12 The rule could be: Final Value = 3 times the Starting Value Final Value = 8 more than the Starting Value Final Value = 8 less than 5 times the Starting Value

20. Open Questions
Several of the questions so far have been open questions.
They have either been open-ended, which means several answers will work as correct solutions.
Or they are open in the approach, which means several approaches will work towards finding correct solutions.
Another benefit of asking open questions is that it is a strategy for differentiation.

21. Open Questions as Differentiation
“The ultimate goal of differentiation is to meet the needs of the varied students. (pg 6)”
Asking open questions makes this more manageable as they allow different students to approach a problem using different strategies.
They also allow students at different levels to benefit from working through the questions at their level.
“Struggling students are less likely to be the passive learners they so often are” (pg 6)
From Small

22. Open Questions
Another benefit… Students who may not normally feel successful in the mathematics classroom CAN!! Students can gain confidence because they can correctly answer your questions the first time!

23. Strategies for making Open Questions
Ask for observations.
Ask for similarities or differences.
Replace a number with a blank.
Turn a question around.
Use “soft” words.
From Small (2012) pg 7

24. Strategies for making Open Questions
Ask for observations. What do you notice about the graph at the right? Line; quadrants 1 & 3; x- axis goes from -5 to 5; y- axis goes from -6 to 6; line goes up from left to right; increasing function; etc…

25. Strategies for making Open Questions
Ask for similarities or differences. How are the numbers 24 and 48… Similar? all numbers are even; 48 is twice 24; 2-digit numbers; 2 twos is 4 & 2 fours is 8; etc. Different? 2 tens vs. 4 tens; 4 ones vs. 8 ones; 24 rounds down to 20 & 48 rounds up to 50; etc.

26. Strategies for making Open Questions
Replace a number with a blank.
Instead of asking:
There are 25 students in one classroom and 31 in another. How many students are there altogether?
You could ask:
There are 56 students total in 2 classrooms. How many students are in each classroom?
From Small pg 8

27. Strategies for making Open Questions
Turn a question around.
Instead of asking:
What is half of 20?
You could ask:
10 is a fraction of a number.
What could the number & fraction be?
From Small pg 8

28. Strategies for making Open Questions
Use “soft” words.
Instead of asking:
Give me two numbers that sum to 100.
You could ask:
Give me two numbers that sum to something close to 100.
From Small pg 8

29. Open-ended Assessment in Math. http://books.heinemann.com/math/
x = 6
Write an equation that involves two different operations and has the solution x = 6.
Open-ended Assessment in Math.

30. Your turn… Pick a topic that you teach.
Think of a standard question that you might ask in class or on a test.
Turn it into an open question.
Share with the group.

31. Resources
Open-ended Assessment in Math.
Schuster, L. and Anderson, N.C. (2005). Good questions for math teaching: Why ask them and what to ask grades 5–8. Math Solutions Publications: Sausalita, CA.
Small, M. (2012). Good questions: Great ways to differentiate mathematics instruction.2nd Ed. Teachers College Press: Columbia University.
Small, M. and Lin, A. (2010). More good questions: Great ways to differentiate secondary mathematics instruction.Teachers College Press: Columbia University.
Sullivan, P. and Lilburn, P. (2002). Good questions for math teaching: Why ask them and what to ask grades K–6. Math Solutions Publications: Sausalito, CA.