Often I wonder how much to explain or define for my students before engaging in the problem solving process. Proponents of sense-making in mathematics classes like Dan Meyer and The Math Forum encourage presenting a perplexing scenario to students and letting them develop the questions to be answered using math. This is a very enticing proposition. Who wouldn’t want a math class which uses the Socratic method to solve problems as a community. I do! Professor Ilana Horn recently wrote a piece investigating the merits of this pedagogical philosophy with other popular options like Doug Lemov’s Teach Like a Champion.
Some students, however, need more scaffolding, language support, cultural background, or skill reinforcement before they are ready to grapple with a truly perplexing situation.
For instance, what if your students view their zone of proximal development much differently than you, as the educator, do? What if the student views every problem as lying in the outer ring, but it truly lies in the middle or inner ring according to your professional opinion? Which leads into my question about problem-based learning. How much do you scaffold for students who need it before you set them free to make sense of a great, perplexing mathematical scenario?
This is a major question for special education math teachers. How much scaffolding is too much so that the process of solving the problem is taken out of the hands of the student? One area where this comes up is when teachers are deciding what order in which to present information to students during the problem solving process. As an example, here is a problem I have been developing in which there are two components. Which of these components should go first in a truly problem-based classroom? Maybe you can help me figure it out!
Component #1: The Video
Component #2: The Situation
Which of these components would you present to your students first? Why?
The question for this scenario could be “how much money would you need to wager in order to guarantee victory if you answered the Final Jeopardy question correctly?”
But since we have not asked the question directly there are many avenues in which this situation can be investigated by a class. If you lead a class with a behaviorist iron-fist then you may not have the pleasure of answering questions that you have not thought of and that are truly engaging to your students, like these from Max Ray of the Math Forum,
What if we all get it right? What if only I get it wrong? What if we all get it wrong? Can I make a wager where I win no matter what?
Great post Andrew! Very thought provoking. I would present component #2 first, but with a different picture, the Final Jeopardy scenario from component #1. Just for the consistency. I’d go with component #2 first because it sets the table for the video that follows. Nothing to solve, nothing to figure out, just the noticing and wondering. Then go to the video. You could follow up later with the other picture, or maybe a series of other similar pictures, and the, “How much would you need to wager…?” question.
One thing that might be interesting is for you to have your kids write their own Final Jeopardy questions. Or maybe 2, one they would consider easy (learner can do unaided) and one they would consider challenging (learner cannot do). You might get an insight into their zones.
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Thanks for the comment Joe!
I think you’re absolutely right about consistency of setting. Students in special education classes often struggle with transfer of learning from one related problem to the next, let alone two different Jeopardy game settings.
I love the assessment task idea! I especially like the meta-cognitive assigning of levels of challenge!
I did read this when you wrote it, but just had a new thought. I don’t think that every scenario you present students has to be perplexing. It can be any situation that involves math that doesn’t include a question. For example, “The corner deli sells roses in bunches of 6. If Dylan buys 3 bunches of roses, how many roses does he have?” often leads to answers of 9, since they see the 6 and 3 and figure adding is a good idea. Instead, ask this: “The corner deli sells roses in bunches of 6. Dylan buys 3 bunches.” What do they notice and wonder? Or even, “The corner deli sells roses in bunches of 6. Dylan buys 3 bunches. Draw a picture of the story.” Anything we can do to help students realize the value and power of doing sense-making will help them become better problem solvers. And it can start with really simple things!
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