Formative Assessment – Leading Learners to Level Up (#LL2LU) – A Definition of Derivative

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

Do we offer learners a way to decode their understanding and ask targeted questions to improve their work? Have we, as a team, identified what is essential to learn and at least one path to success? How do we communicate and collaborate with our learners and our colleagues to build paths to success for all learners?

What if we offered non-graded self-assessment that helped everyone calibrate what they know with what is essential to learn?

What does it take for a calculus learner to show they understand how to use a definition of the derivative? What might the essential learning look like?

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What must one be able to do to show understanding of this topic in calculus? Can the teaching team break it down into components that are necessary to use this definition of a derivative?

In our brainstorming session, Sam and I decided that the following was needed when finding a derivative analytically using this definition:

You have to understand function notation. You have to be able to use function notation when x is a constant and when x is a variable. You have to be able to simplify polynomial and rational expressions.

Do our young learners understand what the words in the previous paragraph mean? Do they need to see it too?  Are we communicating clearly? Is there a difference between saying I understand it and actually understanding?

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What if we asked our learners to show what they know by working through a document to “level” where they are based on our expectations?

Would young learners of calculus be empowered and emboldened to ask specific questions about their understanding? In other words, will learners begin to differentiate for themselves? I assure you it is much more rewarding and fun to be asked specific questions instead of fielding “I don’t know nothing” statements.

Would this type of ungraded self-assessment build confidence and relationship? From experience, I know the answer is yes IF the message is that it does not matter where you are as long as you are working to get to Level 3.  As a community, we must all get to Level 3, the essential learning.

How might we change the conversation in our classes?  How might we use assessment for learning? How might we provide enough insight to compel learners down a path?

The excitement of learning, the compelling personal drive to take one more step on the path towards wisdom, comes when we try to solve a problem we want to solve, when we want to solve, when we see a challenge and say yes, I can meet it.  Great teachers lead us just far enough down a path so we can challenge for ourselves. They provide us just enough insight so we can work toward a solution that makes us, makes me want to jump up and shout out the solution to the world, makes me want to step to the next higher level. Great teachers somehow make us want to ask the questions that they want us to answer, overcome the challenge that they, because they are our teacher, believe we need to overcome. (Lichtman, 20 pag.)


Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.


  1. I like that you’re asking students to assess their own learning — and to take the extra step of “leveling” their current understanding. I think students can get frustrated when they see where they are and where they’re supposed to be without understanding the path between the two. This “step” format (1, 2, 3, 4) helps them not only assess where they are, but it shows them specifically how a 1 can move to a 3, and what comes in between. Even though I don’t remember the calculus involved here, I can see how each of the levels builds on the previous to move a student toward the Level 3 essential learning goal.


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