[LOA] Hypothesis #4: Right Question / Right Rung

#4. Choose the right question for the right rung.

This is a high-level abstraction of cities on a map:

I don’t think it’s easy to start so high up on the ladder and answer questions like:

  1. “Can you guess where they should put the new cell tower?” or
  2. “What information will be important to know here?” or
  3. “How should we represent that information?”

Guessing, it seems to me, is a task that is easier to perform at lower level of abstractions. (Like this one.) Meanwhile, it’s impossible for the student to consider the lower-level question, “What information will be important to know here?” when the important information has already been selected. (The relative locations of the cities.) It’s impossible to consider the question, “How should we represent it?” when the representation has already been selected. (A coordinate plane.)

Likewise, it’s impossible to ask a student to “Calculate the location of the new cell tower” when they’re looking at a low-level abstraction of the context. Calculation is a task that’s made possible by higher levels of abstraction.

Again, we find a limitation of print-based curricula. The authors choose to show a single level of abstraction of a context and then ask all their questions about it, whether or not they’re the right questions for that rung.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. One piece of information that’s important to know is the terrain . If the right configuration of mountains and valleys and ridges are in place, it might be impossible to place a cell phone tower that is visible by all 3 towns.

  2. If the towns were Bayville Seaburry and Medfield, the question would have an idiotic answer. So maybe one should discuss the problem what makes a good placement first.


  3. @R. G.’s response provokes me to realize that we can solve the problem by placing the tower at infinity.

    Let’s look at what we’re doing here:
    1. We found a problem in a textbook that looked promising.
    2. Upon further examination, it was disappointing.
    3. Let’s make it better.

    Engaging in item3 is part of what makes teaching so fun.

  4. > 3. Let’s make it better.
    > Engaging in item3 is part of what makes teaching so fun.

    You’re so right. I wished I could educate students to appreciate this procedure.

    Student: “So what are we expected to do here?”
    Teacher: “Improve!”

  5. The more I read the fantastic ideas you all have, the more I hate myself. I have shamelessly stolen so much off this site for my own teaching. I wish I Gould give back, sorry.