I’m really good at enjoying the cleverness of a scenario and grafting (sometimes seamlessly, sometimes less so) it onto a mathematical standard (or two or three). I’m less good at starting with a standard (or two) and designing a scenario that appropriately and precisely maps onto it. Sometimes that results in a problem that doesn’t – in a targeted way – address the standard I’m hoping students will take away from it. Sometimes I wind up developing four problems that require students to develop a polynomial expression using the same idea without really introducing anything new or extending it. We do a lot of standards mapping and curriculum mapping, but rarely do we do question mapping.
For example, I’ve facilitated and messaged this problem followed by these problems. The scaffolding and teaching (I hope!) will address different standards. But the problems themselves don’t necessarily necessitate different methods or manipulation of polynomials or quadratics.
The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.
In order to achieve this dance, there might be subtle differences in the way a problem is posed. Consider this an attempt to get better at that backwards design approach and to ensure that we’re eliciting the right question.
1. Start with the standard. Hey, here are some standards!
2a. What is the question that you want students to ask that points to the standard?
2b. What might be the language and vocabulary in which students ask it? Because students probably won’t ask “how do we find the roots of a polynomial?”, but they might ask “how do we find where the curve crosses the x-axis?”.
3. What is a possible scenario or task that will elicit that question?
[Optional?] Check your work: Are there other standards that this scenario might address? Are there other ways to solve it that skate around the standard you’re aiming at? Maybe consider giving it a trial run by posing it to a colleague and see if they get close to your intended question?
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OK so let’s try this.
- I’ll pick a standard. How about this one.
CCSS.MATH.CONTENT.HSF.BF.B.5
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
2a. What is the question that you want students to ask that points to the standard?
How do I find the inverse of this here equation that has an exponent (or logarithm) in it?
Perhaps something along the lines of y=ab×.
2b. What might be the language and vocabulary in which students ask it?
How do I find the solution of this here equation that has an exponent in it?
3. What is a possible scenario that will elicit that question?
Me thinking: Well there are lots of applications of things with exponential growth and decay. Populations, investments, radiation and half-life. Perhaps a solicitation letter asking students to analyze bacterial growth of a certain strain?
Or maybe we go abstract and posit something like:
What is the intersection of these two functions? (Or “what do you notice and wonder?”)
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I’d also suggest that the practice of Question Mapping might actually help in facilitation as well? Namely that you have a question in the back of your pocket that you know you need to get the students to ask. And if they’re not asking it you need to pull it out of them with leading questions or other bread crumbs. For the problem draft above, I’m not moving on until we establish the questions in 2a and 2b as the impetus for the lesson.
It might be fun (and enlightening) to have a curriculum map of questions along with your standards. And shoot, you’d have your semester review already written months in advance.
emergent math 
When you are going through all of these pieces for mapping it makes life much easier. I have found really going backward and going from standards to tasks. It is really change is some topics but in most situations it is valuable to take standards and look at what we want students to get and take with them. It really allows for education to have purpose and guidance for every level.
When you question how do you come up with the valuable questions. Questioning is an area that I struggle. Is there something that I can do to start with when coming up with questioning?
When you are going through all of these pieces for mapping it makes life much easier. I have found really going backward and going from standards to tasks. It is really change is some topics but in most situations it is valuable to take standards and look at what we want students to get and take with them. It really allows for education to have purpose and guidance for every level.
When you question how do you come up with the valuable questions. Questioning is an area that I struggle. Is there something that I can do to start with when coming up with questioning?
Hi Becky, I suppose there are the in-the-moment questions and the Big Question that’ll address the standard. While this post mainly focuses on the Big Question, it’s probably helpful to consider it to get at those smaller, more probing questions.
Consider: what’s the Big Question? And what to students already know about the topic. That might help you pre-plan the smaller questions that will push for deeper thinking.
The Shell Formative Assessment Lessons (http://map.mathshell.org/lessons.php) have some great examples of those types of questions in their pre-assessment facilitation notes.
Hope this helps!
–geoff
I find myself struggling at times to nail a single task down to a grade level or standard when creating problem based lessons….it’s tough. I always worry that if I focus on the standard first when creating tasks, the initial question won’t come naturally. I just don’t want the question/task to feel forced. Thanks for the emphasis on purpose and clarity.