Getting Students to Ask Better Questions

Avery Pickford5th & 6th Grade Math Teacher

The Nueva Schoolavery.pickford@gmail.com

@woutgeo

How many points would you give to this 3rd grade student if the problem

is worth 5 points?

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0

1

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5

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41

37

40

Votes

Votes

not fair

Questioning:

A Pretty Great Way to Get A Sense of What Students May Know

Terdragon Curve Found only in the boundary of the Rauzy fractal in the Hausdorff dimension, the terdragon curve is a large, vocal gasket with a distinctive tail with alternating black and white rings.

Male and female terdragon curves are similar physically. They are roughly the same size, measuring about 42.5 sm from head to rump.

Highly sociable, terdragon curves live in groups averaging 17 members.

Ring Tailed LemurFound only in the southern part of Madagascar in the dry forest and bush, the ring-tailed lemur is a large, vocal primate with a distinctive tail with alternating black and white rings.

Male and female ring-tailed lemurs are similar physically. They are roughly the same size, measuring about 42.5 cm from head to rump.

Highly sociable, ring-tailed lemurs live in groups averaging 17 members.

Found only in the southern part of Madagascar in the dry forest and bush, the ring-tailed lemur is a large, vocal primate with a distinctive tail with alternating black and white rings.

Male and female ring-tailed lemurs are similar physically. They are roughly the same size, measuring about 42.5 cm from head to rump.

Highly sociable, ring-tailed lemurs live in groups averaging 17 members.

Types of Questions

Administrative-questions that aid the

running of a classroom

1. Where is your homework?2. Why didn’t you take notes on

my lecture on terdragon curves?

Types of Questions

Factual- questions that gatherinformation

1. What is the definition of prime?

2. Where are terdragon curves located?

Administrative

Types of Questions

Clarifying- questions concerning

rules and/or assumptions

1. Can a negative number be prime?

2. Can terdragon curves move?

Administrative

Factual

Types of Questions

Procedural- questions concerning

how to do something

1. What is the 7th prime number?

2. How long would 4 terdragon curves be?

Administrative

Factual

Clarifying

Types of Questions

Probing-questions that ask

students to articulate or elaborate ideas

1. How would you prove that 43 is prime?

2. Describe the Hausdorff dimension.

Administrative

Factual

Clarifying

Procedural

Types of Questions

Conceptual-questions that explore

underlying mathematical reasoning and connections

1. Why did mathematicians choose to make 1 non-prime?

2. Why do terdragon curves have tails with alternating black and white rings?

Administrative

Factual

Clarifying

Procedural

Probing

Types of Questions

Divergent-questions that vary and

extend

1. What if primes were defined as whole numbers with exactly 3 factors?

2. What would happen if we moved a terdragon curve outside the Rauzy fractal?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Types of Questions

Contextual-questions that link to

other mathematical and non- mathematical content

1. When are primes used outside of math class?

2. Why are lemurs so cute?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Types of Questions

Consensus Building-questions that motivate a

group understanding

1. Do you agree or disagree with Arsha’s statement that 2 is not prime because it is even?

2. Can you restate what Bob said about terdragron curves in your own words?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Does that make sense?

rhetorical or probing?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Levels of Demand

• Memorization

• Procedures without connections

• Procedures with connections

• Doing mathematics

Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics teaching in the middle school, 3(4), 268-275.

The Tasks We Give

“Not all tasks provide the same opportunities for student thinking and learning.”

Hiebert, J. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Pickford, A. (2015). My opinion on stuff. Palm Spring, CA: Unpublished.

A 5th grader is going to ask different questions…

You are planning your birthday party and you want people to show up early. So you decide to tell people that whoever shows up first gets half your cake, whoever shows up next gets half of what is left, and so on. How much of the cake is left after the 5th person shows up?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Are these the same problems?

You are planning your birthday party and you want people to show up early. So you decide to tell people that whoever shows up first gets half your cake, whoever shows up next gets half of what is left, and so on. How much of the cake is left after the 5th person shows up?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Can I go to the bathroom?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

What’s a monomial?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

In the Birthday Cake problem, do you get a piece of cake?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

How do I divide fractions?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Why does that work?Does that always work?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Why can I just “flip and multiply” when dividing fractions?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Ah! I have Taylor Swift’s “Shake it Off” stuck in my head! Could we create a metric to measure “song catchiness?”

Habit of Mind:Tinker and Invent

Makes changes in order to solve existing problems• Looks at simpler examples when necessary• Looks at more complicated examples when necessary/interesting • Varies parameters in regular and useful ways

Create new problems• Creates variations: problems/questions that explore a different case

(operation, number, etc.) from the original• Creates generalizations: problems/questions that explore a group of

cases at the same time.• Creates extensions: new problems/questions (that aren’t variations or

generalizations) that are inspired by the original problem.  These may involve changing the constraints (rules) of the original problem.

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

When will we ever use this?

What types of questions are students asking?

Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Who did this in a different way?Do you agree or disagree with that?

A Culture of Inquiry: Cups

A Culture of Inquiry:Skeptical Peers

a) Find a number that has exactly 4 factors.

b) Find a number that has exactly 5 factors.

c) Find a number that has exactly 1 factor.

What types of questions would you expect from

students?Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

The locker Problem: Imagine you are at a school with 100 students and 100 lockers in one long row, all shut and unlocked (the lockers, not the students). Suppose the first student goes along the row and opens every locker. The second student then goes along and shuts every other locker beginning with locker number 2. The third student changes the state of every third locker beginning with locker number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.) The fourth student changes the state of every fourth locker beginning with locker number 4. Imagine that this continues until all 100 students have followed the pattern with the 100 lockers. At the end, which lockers will be open and which will be closed? Why?

What types of questions would you expect from

students?Administrative

Factual

Clarifying

Procedural

Probing

Conceptual

Divergent

Contextual

Consensus Building

Habits of Mind:Pattern Sniffing

• Square lockers are open• Numbers with an odd number of factors

are open• Once student #5 is finished, lockers 1-5

won’t be touched again• The number of closed lockers in a row

form the sequence 2, 4, 6, 8, …• All prime number lockers will be closed

Why?

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Getting Students to Ask Better Questions

Avery Pickford5th & 6th Grade Math Teacher

The Nueva Schoolavery.pickford@gmail.com

@woutgeo