Reflection: Grappling with Complexity Which Way Do We Go? - Section 3: Notes


I have found that moments of surprise are incredibly powerful learning moments. When students predict one thing, but something else happens it seems to emphasize the moment and create instant engagement. It causes students to focus in and heightens their curiosity and need to know why. It also seems to anchor the learning more strongly in their minds and make it more memorable, and hopefully easier to recall.

I try to use what I call "predict & surprise" as much as I can. Its easiest to plan these experiences around common misconceptions or ideas that tend to be tricky for students. In this lesson I expect students to be surprised when they find that integers don't behave in a way that is consistent with their past experience. Operations with integers seem to contradict students' prior knowledge with whole numbers (adding makes numbers bigger and subtracting makes them smaller, etc.)

Working specifically with a number line can help them make sense of what's happening when they notice that:

  • adding negatives and adding positives move in opposite directions and that moves in both directions cancel each other out.
  • adding positives and subtracting negatives move in the same direction.

The number line helps students model the surprising "behavior" of integers so they can visualize it. Students also learn to use the number line as evidence for explaining their results, especially when they use it in conjunction with context that helps provide meaning for the moves.

Once my students have experienced a surprise they are hooked and it is really important to follow up with sense making. In this case I ask questions like:

  • "Why does it make sense that this suprising thing happens?"
  • "How can a number get bigger after subtracting?"
  • "When is subtracting more like adding?"
  • "When does adding act more like subtracting?"
  • "Where do we see that on the number line?" 
  • "How do hot and cold cubes help us understand what's happening in the math?"
  • "What other situations could help us see this?"

Even though my students have 'seen' some of these ideas in previous lessons they need multiple experiences to concrete the learning. This is especially true when they are making such a paradigm shift in math.


  Why surprise helps students learn
  Grappling with Complexity: Why surprise helps students learn
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Which Way Do We Go?

Unit 4: Operations with Integers
Lesson 5 of 24

Objective: SWBAT use a number line to solve integer addition and subtraction.

Big Idea: Students will determine which direction to go on a number line by using the context of hot and cold cubes.

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Math, Number lines , Number Sense and Operations, Operations and Expressions, negative number, integer addition & subtracti, cognitive conflict, cognitive surprise
  54 minutes
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