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)

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 3 grade students: rd -
 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.

To remove barriers to learning while still challenging each student to take risks and responsibility for learning
(Karp & Howell, 2004)

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 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.

 Focus on the Big Ideas of Mathematics
 Student Choice
 Pre-assessment

 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 90 0 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 cm 2 .
What might it be?

Parallel Tasks
Common Follow-up questions could be:
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Is your object big or not so big? How did you know?
Could you hold it in your hand?
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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?
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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.

 Open Questions and Parallel Tasks
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Created by:
Lesson Goal:
Standard(s) Addressed:
Underlying Big Idea(s):
Open Question(s):
Parallel Tasks:
Option 1:
Option 2:
Date:
Grade Level:
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.