Getting Students to Ask Better Questions Avery Pickford 5 th & 6 th Grade Math Teacher The Nueva

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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 Lemur 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. 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 gather information 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 7 th 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),

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 5 th 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 5 th 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 5 th 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 Pickford 5 th & 6 th Grade Math Teacher The Nueva