Enriching Mathematics
Instruction
Opening up our Questions
TLQP 2013-14
Thomas F. Sweeney, Ph.D
The Sage Colleges
Today’s Goals:
•Classify open and closed problems and tasks;
•Understand how open ended problems
promote student learning;
•Generate open-ended tasks from closed ones;
•Learn the importance of good questioning.
CCSSM

New level of rigor
Deeper understanding

A broadening in our teaching styles
APPR is attempting to measure this
Why are the questions we ask our
students so important?
Promotes higher level thinking
Facilitates productive discussion in the
classroom
Students are better able to make sense of
ideas, demonstrate understanding, and reflect
on their thinking.
Successful implementation of CCSSM 
Transition to:
students working in small groups to solve
open-ended mathematics problems
The problem tasks should provide:
(i) opportunities for students to generate several
options and solutions;
(ii) opportunities for students to discuss together;
(iii) opportunities for students to make decisions
and justify their decisions.
Recall: Border Problem
Traditionally “closed” questions :
•involve recalling a fact or performing a skill;
•a single answer;
•content specific - often routine;
•a prescribed solution technique.
Features of Open-ended Problems:
• No fixed method
• No fixed answer/many possible answers
• Solved in different ways and on different
levels (accessible to mixed abilities).
• Offer pupils room for own decision making
and natural mathematical way of thinking.
• Develop reasoning & communication skills
Open-ended task/ problem
They can include tasks that ask
students to explain, justify, and make
predictions beyond the information
they are given.
Good Questions Are Self-Differentiating
Many questions worth asking can be answered
on various levels according to a student's
conceptual understanding.
Thus, they are self-differentiating.
Students with stronger skills or a deeper
understanding can probe the question in depth
or uncover all the possible answers, while other
students might respond on a more basic level.
The teacher also can prepare one or more
follow-up questions to extend the challenge.
Examples
Closed question: "What are the factors of 36?“
Open question:
"Find a number under 100 that has lots of factors."
Closed question:
Find the mean of 8, 10, 12, 12, and 18.
corresponding Open-ended question :
The mean of a set of 5 scores is 12. 'What
might be the scores?
Closed:
What is the area of a rectangle 8 meters long
and 3 meters wide?
Open:
A rectangle has a perimeter of 30 m. What
might be the area? Explain.
Closed- Ended
Find the value of n if
3 x n = 12
Open-Ended
Solve the riddle using the clues and numbers in the table.
40
4
14 12
59 37
1. It is not 3 x 4.
2. It is not > 56.
3. It does not equal 4 tens.
4. It does not equal 2 x 7.
5. It is not the missing number
in 3 x n = 12.
The number is ______.
closed
I want to paint the wall but not the
door. What area of wall do I need to
paint?
open
The area of the shaded part of this
diagram is 8m2• What might be the
dimensions of the large and small
rectangles (Give at least 3 answers)
Closed question:
What is 8 x 6?
Open question:
Suppose you forgot what 8 x 6 is, but
you remembered that 5 x 6 is 30.
How could you use this fact to
figure out what 8 x 6 is?
Why open-ended tasks?
Open-ended questions help students relate
math to the real world.
Students are often required to make decisions
and be creative in finding ways to answer the
questions they are given.
When teachers can assess all these things they
can be sure that the students really understand
the mathematics they are being taught.
Why open-ended tasks?
Students might be asked to bring together and
synthesize a range of mathematical concepts and
skills in solving one problem.
The question might trigger a student-driven
investigation or exploration.
The teacher learns about the students'
conceptual understanding and problem-solving
skills.
Advantages
1). Students participate more actively in lessons
and express their Ideas more frequently.
2). Students have more opportunities to make
comprehensive use of their mathematical
knowledge and skills.
3). Every student can respond to the problem in
some significant ways of his / her own.
4) Teachers can better assess students
understanding and misconceptions.
What Makes a Task Rich?
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•
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Significant mathematics
Mathematical Practices
Multiple layers of complexity
Multiple entry points
Multiple solutions and/or strategies
Leads to discussion or other questions
Students are the workers and the decision
makers
• Warrants reflection - Paired with discourse
Making questions open ended
• Method 1: Working backwards
1. Indentify a mathematical topic or
concept.
2. Think of a closed question and write
down the answer.
3. Make up a new question that
includes (or addresses) the answer.
Method 1: Working backwards
Example:
1. Multiplication
2. 40 x 9 = 360
3. Two whole numbers multiply to
make 360. What might the two
numbers be?
Method 1: Working backwards
Round this decimal to one decimal place:
5.7347
can become ….
A number has been rounded off to 5.8.
What might the number be?
Method 1: Working backwards
Find the difference between 6 and 1
can become ….
Method 1: Working backwards
The difference between two numbers
is 5.
What might the two numbers be?
Measurement Example
• Outcome – Recognize and demonstrate that objects
of the same area can have different perimeters.
• Typical Question (closed task, no choice):
Build each of the following shapes with your color
tiles. Find the perimeter of each shape.
Which shape has the greatest perimeter?
Measurement Example (continued)
New Task (open, choice in number of tiles):
Using 8, 16, or 20 color tiles create different shapes and
determine the perimeter of each. Record your findings on
grid paper.
–
–
–
–
What do you think is the smallest perimeter you can make?
What do you think is the greatest perimeter you can make?
Prepare a poster presentation to show your results.
Sides of squares must match up exactly.
Allowed
Not Allowed
Your turn.
Compute .
Open up the following.
Making questions open ended
Method 2: Adapting a standard question
Indentify a mathematical topic or
concept.
Think of a standard question
Adapt it to make an open ended
question.
Method 2: Adapting a standard question
What is the time shown on this clock?
Can become…
My friend was sitting in class and she
looked up at the clock. What time might
it have shown? Show this time on a clock
Method 2: Adapting a standard question
731 – 256 =
Can become…
Arrange the digits so that the
difference is between 100 and 200.
Method 2: Adapting a standard question
Ten birds were in a tree. Six flew away. How
many were left?
Can become…
Method 2: Adapting a standard question
Ten birds were in a tree. Some flew
away. How many flew away and how
many were left in the tree?
Method 2: Adapting a standard question
Your turn.
Open up the following.
What are the next three numbers in the
following sequence?
1, 4, 7, 10, 13, ___, ____, ____
Method 3:
Ask Students to Create a Situation or an
Example That Satisfies Certain Conditions
Questions of this type require students to
recognize the defining characteristics of the
underlying concept. Students must take what
they know about a concept and apply it to
create an example.
Sample Elementary-Level Questions
Make a 4-digit even number using the digits
below. Explain why your number is even.
3 6 7 1 5
_____ _____ _____ _____
Give an example of an event that has a
probability of 0. Explain how you know the
probability is 0.
Draw a rectangle and label the sides so
that the perimeter is between 19 and 20
units. Explain how you know the perimeter
is greater than 19 and less than 20.
Draw a triangle that has the line of
symmetry below.
Sample Middle School-Level Questions
Identify three numbers whose greatest common
factor is 5 and whose least common multiple is
180. Describe how you found the numbers.
Create a set of data that would satisfy the
following conditions:
The set includes 7 data points.
The range is 10 units.
The mean is greater than the median.
Show that your data set satisfies the conditions.
Fill in values for a and b to make the
equation below true. Explain why your
equation is true.
Draw a quadrilateral ABCD that has one and
only one line of symmetry. Explain why your
quadrilateral satisfies the given condition.
Method 4:
Ask Students to Explain Who
Is Correct and Why
These types of items present two or more
views of some mathematical concept or
principle and the student has to decide
which of the positions is correct and why.
Sample Elementary-Level Question
Of the coins made by the U.S. Mint in one year,
73% were pennies and 6% were quarters.
Suppose you could have all of the coins of one
type. Alex says you would get more money if
you had all the pennies. Austin says you would
get more money if you had all the quarters.
Jenna says it depends on how many of the two
coins were made. Who is right and why?
Sample Middle School-Level Questions
Casey claims he has divided the rectangle
below into four equal areas.
Terrell disagrees.
Who is correct and why?
Creating Open Ended Questions
from Closed Ended Questions
But first . . .
When you create open-ended items, make sure they
are really different from traditional items. For example,
the following items is really no different than simply
asking students to solve the equation:
Johnny solved 2x + 4 = 8 and got 2. Susie solved the
equation and got 6. Who is correct and why?
Rani said the solution to the inequality
(x - 2)(x - 4)< 0 is 2<x<4.
Susie solved the inequality and got x<2 or x>4.
Who is correct and why?
The created item should require students to explain
their reasoning, not simply to reproduce an algorithm.
Open up the questions on the handout ;
or
Create several open questions of your own.
Probably best to work by grade level
Some important considerations
•The mathematical focus
•The clarity of the task / question
•That it is open ended
Thoughts from Teachers
Singapore Teacher Comments
Teacher X: “A lesson such as an open-ended problem
solving was indeed a fruitful one for my pupils and
me. There was another avenue to motivate pupils to
learn mathematics. ….When the pupils got
started to work in pairs, I could see that they were
thinking very hard as to how the work could be
done. I heard comments like’ How can a person be so
heavy?’ and ‘It is not possible to have so many
passengers on a bus?’. I was glad that they tried to
explain and justify to their peers…” (primary four)
Teacher Y: “From my pupils’ work, I feel that
critical thinking can be taught. These pupils were
frightened at first to take the first step, for fear of
getting the answers wrong. But after seeing their
friends’ answers and reasons, these fears were
lifted…..Lastly, the pupils do include their own
personal experiences and knowledge. I was quite
impressed.” (primary four)
Teacher Z: “ ….Another advantage of open-ended
investigation is that learning is more active.
Pupils had a great time making wild guesses, trying to
defend their procedure. However, our pupils are still
not accustomed to accepting other pupils’ alternative
answers, it will take them some time to recognize the
fact that there are sometimes more than one solution
to a problem…… Open-ended investigations here no
doubt require more preparation and time on the part
of the teachers, but judging from the positive learning
outcome of the activity carried out from most of the
pupils, the pupils will begin to see mathematics can be
meaningful and interesting. (primary six)
What do you think?
• When: When would you provide students
opportunities to engage in open or parallel questions?
– near the beginning or end of a lesson or unit?
– before or after closed problems?
• How: How would you use open and parallel tasks?
– assessment for, as or of learning?
– in groups or individually?
• Conditions: What classroom conditions are needed to
be to be successful engaging in open and parallel
tasks? How would you build this capacity?
Building open ended tasks into a lesson
It is important to plan two further
questions/ prompts:
•For those children who are unable to start
working (enabling prompts).
•For those children who finish quickly
(extending prompts).
Why are the questions we ask our
students so important?
• Promotes higher level thinking
• Facilitates productive discussion in
the classroom
• Students are better able to make
sense of ideas, demonstrate
understanding, and reflect on their
thinking.
Generic Set of Questions?
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•
•
•
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?
• Jake, 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 (the teacher)
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
Strategies to
Improve Questioning
Plan questions in lesson design
•Choose a variety of questions
•Video some lessons to assess level of questioning
•Focus questions on student understanding;
remove focus from right/wrong answers.
•Assume all answers are meaningful.
•Allow multiple opportunities for interaction
centered around math ideas: questioning,
discussion, and reflection.
•Increase wait time.
•
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…
References:
Designing open-ended question Louise Hodgson 2012
http://pt.slideshare.net/evat71/designing-quality-open-ended-tasks