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
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.

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.
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?
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?
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?
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…
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
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
Asking Questions

1. Ask open ended, not just close ended
questions…
 2. Ask divergent as well as convergent
questions…
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…
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…

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
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?
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.
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________.
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.
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?
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?
Measurement

Weight
 Area
 Length and Perimeter
 Volume and Capacity
Weight

I went shopping and found a one pound box
of Snowflake Sugar Cubes for $1.70. A 1kilogram box of White Cloud Sugar Cubes
offered a better buy. How much might the
White Cloud Sugar Cubes have cost?
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.
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?
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?
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?
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?
Length and Perimeter

A television screen measures 41 inches
along its diagonal. What might the length
and width of the screen be?
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?
Volume and Capacity

Could $1 million fit into a standard-size
briefcase? Assume the largest denomination
in circulation is $100.
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?
Geometry

Using your protractor, can you draw three
adjacent angles that show at least one acute,
one right, one obtuse and one reflex angle?
Geometry

Find a quadrilateral that can be inscribed in
a circle and one that cannot
Geometry

It is possible to inscribe only parallelograms
that are also rectangles
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.
Geometry

This question will help students understand
that all congruent shapes are also similar
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.
Geometry

Do students understand that the length of
the lateral surface must be the same as the
circumference of the base?
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?
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?
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.
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…
The Phantom Tollbooth
Perhaps 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)
