Transcript Differentiating with Questioning

The Research of Dr. Marian Small
Disseminated through the following books:
Good Questions - Great Ways to Differentiate Mathematics Instruction
More Good Questions - Great Ways to Differentiate Mathematics Instruction
And summarized in the following slides.
Equity
Equity does not mean that every student
should receive identical instruction;
instead it demands that reasonable and
appropriate accommodations be made as
needed to promote access and attainment
for all students
(NCTM, 2000, p.12)
Student Approaches
One way that we see the differences in students is
through their responses to the mathematical questions
and problems that are put to them. For example,
consider the task below, which might be asked of 3rdgrade students:
 In one cupboard, you have three shelves with
five boxes on each shelf. There are three of those
cupboards in the room. How many boxes are
stored in all three cupboards?
How might students approach this task?
Student Approaches
Liam immediately raises his hand and simply
waits for the teacher to help him.
Angelita draws a picture of the cupboards, the
shelves, and the boxes and counts each box.
Tara uses addition and writes 5 + 5 + 5 + 5 + 5 + 5 +
5 + 5 + 5.
Dejohn uses addition and writes 5 + 5 + 5 = 15, then
adds again, writing 15 + 15 + 15 = 45.
Rebecca uses a combination of multiplication and
addition and writes 3 x 5 = 15, then 15 + 15 + 15 =
45.
The Goal of Differentiating with
Questioning
To remove barriers to learning while still
challenging each student to take risks and
responsibility for learning
(Karp & Howell, 2004)
Zone of Proximal Development
One approach to meeting each student’s
needs is to provide tasks within each
student’s zone of proximal
development and to ensure that each
student in the class has the opportunity
to make meaningful contribution to the
class community of learners.
Zone of Proximal Development
Zone of proximal development is a term
used to describe the “distance between
the actual development level as
determined by independent problem
solving and the level of potential
development as determined through
problem solving under adult guidance or
in collaboration with more capable peers.”
(Vygotsky, 1978, p. 86)
In Other Words . . .
In other words, working within a student’s
zone of proximal development is to
assist students in accessing new ideas
that are close enough to what they
already know to make the access feasible.
Elements of Effectively
Differentiated Instruction
Focus on the Big Ideas of Mathematics
 Student Choice
 Pre-assessment

Core Strategies for Differentiating
Mathematics Instruction
Open Questions
Questions framed in such as way that a
variety of responses or approaches are
possible
 Parallel Tasks
Sets of tasks, usually two or three, that are
designed to meet the needs of students at
different developmental levels but that
get at the same idea and can be discussed
simultaneously

Open Questions
The same question is put to the entire class
but it is designed for differentiation of
response based on each student’s
understanding.
Open questions need just the right amount
of ambiguity.
Any question too specific may target a
narrow level of understanding.
Open Questions
For example, instead of asking students to
find 25% of 48 . . .
Ask students to fill in values for the blanks
to make this statement true:
72 is _____% of _____
Open Questions
1. The number 4 is a factor of two different
numbers. What else might be true about
both of the numbers?
2. An expression involving the variable k
has the value 10 when k=4. What could
the expression be?
3. The mean of a set of numbers is 8. What
might the numbers be?
4. A shape has six sides and two 900 angles.
What could it look like?
Parallel Tasks
To create parallel tasks to address a
particular Big Idea, it is important to
first think about how students might
differ developmentally in approaching
that idea/concept.
Then the object is to develop similar enough
contexts for the various options that
common questions can be asked of the
students as they reflect on their work.
Parallel Tasks
For example, if the Big Idea is
Measurement, a set of Parallel Tasks
might be as simple as letting students
chose whether they are comfortable with
linear or area measurements:
Option 1 An object has a length of 30 cm.
What might it be?
Option 2 An object has an area of 30 cm2.
What might it be?
Parallel Tasks
Common Follow-up questions could be:
 Is your object big or not so big? How did
you know?
 Could you hold it in your hand?
 How do you know that your object has a
measure of about 30?
 How would you measure to see how close to
30 it might be?
 How do you know that there are a lot of
possible objects?
Parallel Tasks
To use and take advantage of the
relationships between the operations in
computational situations
Option 1 Show that the product of two
numbers can sometimes be greater than
the quotient and sometimes less
Option 2 Choose two numbers to make
the following statement true:
quotient < difference < sum < product
Parallel Tasks
Questions that could be asked of both
groups include:
 Suppose you were using 25 and 5. Which
is greater – the quotient or the product?
 Is it ever possible for the quotient of two
numbers to be greater than the product?
If so, when might that be? If not, why
not?
 What two numbers did you choose?
Why did you try those?
Parallel Tasks
An experimental probability approaches a
theoretical one when enough random
samples are used.
Option 1 You roll two dice. Is it more
likely that the sum is 8 or that the
difference is 2?
Option 2 You roll two dice. You want an
event that is only a bit less likely than
rolling a difference of 2. What could it
be?
Parallel Tasks
Questions that could be asked of both groups
include:
 How likely is it that the two rolls of a die
differ by 2?
 What would the probability of an event
need to be for it to be greater than the
probability that the rolls differ by 2?
 Is the probability of rolling a sum of 8 less
than the probability of rolling a difference
of 2?
 How did you solve the problem?
Your Task!
To use Dr. Small’s template for developing
Open Questions and Parallel Tasks and
find or develop at least four Open
Questions and two Parallel Tasks from
your first CMP2 unit.
We’ll then ask you to share your favorite
Question/Task with the entire group.
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Open Questions and Parallel Tasks
Created by:
Date:
Lesson Goal:
Grade Level:
Standard(s) Addressed:
Underlying Big Idea(s):
Open Question(s):
Parallel Tasks:
Option 1:
Option 2:
Principles to Keep in Mind:
All open questions must allow for correct responses at a variety of levels.
Parallel tasks need to be created with variations that allow struggling students to be successful and
proficient students to be challenged.
Questions and tasks should be constructed in such a way that will allow all students to participate
together in follow-up discussions.