The Art of Asking Good Questions

1. The Art of Asking Good Questions • Maria Mitchell • Based on Good Questions for Math Teaching Why Ask Them and What to Ask • By Lainie Schuster and Nancy Canavan Anderson

2. What are Good Questions? They help students make sense of the mathematics. They are open-ended, whether in answer or approach. There may be multiple answers ore multiple approaches. They empower students to unravel their misconceptions.

3. What are Good Questions? They not only require the application of facts and figures but encourage students to make connections and generalizations. 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 investigate this newly found interest.

4. How are Good Questions Created? What are the mathematical goals or objectives? What misconceptions do students have on the content? What connections do you want to make between lesson objectives and previous concepts or facts learned? Assessment of their understandng?

5. Generic Set of Questions? Why do you think that? How did you know to try that strategy? How do you know you have an answer? Will this work for every number, every situation? When will this strategy not work? Can you provide me with a counterexample?

6. Generic Set of Questions? John, do you have a different strategy? How is your answer alike or different from Alyson’s? Can you repeat Brian’s idea in your own words? Do you agree or disagree with Courtney’s idea? Why?

7. Answering and Asking Questions – Students Asking What are some things that you can do when asked a question other than directly answering it? Think of ways that you can help students answer their own questions…

8. Answering and Asking Questions – Students Asking 1. Repeat the question, paraphrasing it… 2. Redirect the question 3. Ask probing questions 4. Promote a discussion among the students

9. Answering Questions 1. Directly answer the question 2. Postpone answering the question 3. Discourage inappropriate questions 4. Admit when you do not know the answer

10. Asking Questions 1. Ask open ended, not just close ended questions… 2. Ask divergent as well as convergent questions…

11. Pauses and Silence 1. Wait, pauses and silence are appropriate class behaviors… 2. Wait, give the students time to think… 3. Wait, or you will establish an undesirable norm…

12. Create an Accepting Atmosphere 1. Ask for questions… 2. Answer questions… 3. Answer student questions adequately… 4. Listen to the question or to any student comments… 5. Do not put down the student…

13. Good Questions to Use in Math Class Number relationships Multiplication and Proportional Reasoning Fractions, Decimals, and Percents Geometry Measurement Algebraic Thinking Data Analysis and Probability

14. Number Relationships 1. What is the smallest numbers that has four and six as factors? 2. How do you know when you have found all possible factors for a given number? What is the greatest factor possible for any whole number? 3. Is zero an even or odd number?

15. Multiplication and Proportional Reasoning Write a story problem for 127/5. Solve your story problem. Does your answer have a reminder? How do you know? If there is a remainder what are you going to do with it? Approximately 11% of the population is left handed. Use this information to estimate the total number of lefties in this school.

16. Fractions, Decimals and Percents Who am I? I am less than one half. I am greater than one third. My denominator is a multiple of three. I am simplified. I am________.

17. Fractions, Decimals and Percents Write a story problem that can be solved with this number sentence? 2 ¼ + 1/8 + 1 ½ = 3 7/8 Justify its solution as well by showing a visual model.

18. Algebraic Thinking The following sign is posted at the Central Library: FINE POLICY FOR OVERDUE BOOKS Twenty five cents per day plus an additional $.50 for reshelving. Maximum fine of $5.00. What might a table or graph look like that the library could also display to help people calculate their fines?

19. Algebraic Thinking Carla and Fiona are having a mathematical debate about the equation y = ½ x + 3. Carla thinks that every time y changes by 1, x changes by two. Fiona thinks that every time y changes by ½, x changes by one. What do you think?

20. Measurement • Weight • Area • Length and Perimeter • Volume and Capacity

21. Weight • I went shopping and found a one pound box of Snowflake Sugar Cubes for $1.70. A 1-kilogram box of White Cloud Sugar Cubes offered a better buy. How much might the White Cloud Sugar Cubes have cost?

22. Weight • Give the students the conversion (1 pound = 0.454 kilograms) so that they can estimate a possible price. Students should realize that 1 pound is about the same as half of 1 kilogram.

23. Weight • Sheila emptied her piggy bank, wrapped her coins, and put them in a bag to go to the bank. The bag weighed 20 pounds. How much money do you think Sheila had in coins?

24. Area • A ball of dough is rolled out into a circle with a 12 inch diameter. How many cookies with a diameter of 2.5 inches can be made from this dough?

25. Area • I wrapped a rectangular box in a piece of wrapping paper that was 11 inches by 17 inches. I had no paper left over but did have some minor overlaps. What might have been the dimensions of my box?

26. Length and Perimeter • Juana is 5.6 feet tall and Jeremy is 5.8 feet tall. Liza is taller than Juana but shorter than Jeremy. How tall might Liza be in feet and inches?

27. Length and Perimeter • A television screen measures 41 inches along its diagonal. What might the length and width of the screen be?

28. Length and Perimeter • A fifteen foot ladder has a warning that reads “Bottom of ladder should rest between 3 and 4 feet from the wall.” What are some heights that a person standing on the ladder can reach?

29. Volume and Capacity • Could $1 million fit into a standard-size briefcase? Assume the largest denomination in circulation is $100.

30. Volume and Capacity • You write on paper every day in school. If you were to box up all of the pieces of paper you’ve used since entering school, what fraction of the room do you think the boxes would fill?

31. Geometry • Using your protractor, can you draw three adjacent angles that show at least one acute, one right, one obtuse and one reflex angle?

32. Geometry • Find a quadrilateral that can be inscribed in a circle and one that cannot

33. Geometry • It is possible to inscribe only parallelograms that are also rectangles

34. Geometry • I drew a set of seven triangles. Five were similar to one another, three were congruent, and two were neither similar nor congruent to any other. What might these triangles have looked like? Draw the triangles and label their side lengths.

35. Geometry • This question will help students understand that all congruent shapes are also similar

36. Geometry • Draw and label the dimensions of a flat pattern for a cylinder. Label the radius of the base and the length and width of the lateral surface.

37. Geometry • Do students understand that the length of the lateral surface must be the same as the circumference of the base?

38. Data Analysis and Probability Make a data set representing the ages of students with the following statistical landmarks: Sample size: 12 students Range: 8 years Median age: 12.5 years Mode: 10 years Will everyone’s data set look the same?

39. Data Analysis and Probability The probability of a particular event happening is 2/5. Explain the probability of the event not happening. What could the event be?

40. Data Analysis and Probability Design a spinner with five spaces so that the chance of landing in one space is twice the chance of landing in each of the other four spaces. Give the degree measurement of each central angle.

41. To Summarize The power of questioning is in the answering. As teacher s we need to ask good questions to promote thinking and get good answers…

42. The Phantom TollboothPerhaps the Dodecahedron says it all best: “That’s absurd,” objected Milo, whose head was spinning from all the numbers and questions. “That may be true,” the Dodecahedron acknowledged, “but if it’s completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you’ll have to make it yourself.” (1961,175)