[AnE] Open question + closed questions

In Mathematics Assessment, I believe there is a real need for a large variety of "open" questions.  But let me elaborate on what I mean here before I get tomatoes thrown at me!

Open questions are questions that are "framed in such a way that a variety of responses or approaches are possible." (Small and Lin 2010)

And that's exactly what it is.  Closed questions are questions that only lead to one answer - and most of the time the skills that they test are quite limited.  Questions like the following examples would be open questions since it provides a wide variety of ways of answering it:

Open question:
Example 1:
Two lines are perpendicular to each other and intersect at the point (10, 5).  What might be the slopes of these lines?  Explain your thinking.

Example 2:
Write an algebraic expression involving one variable and at least three terms that can be simplified by using exponent principles.


On the other hand, the following would be closed questions since they allow for limited approaches toward finding the answer.

Closed question
Example 1:
Determine the derivative of the following function: f(x) = 3x + 10

Example 2:
Determine the slope of the line connecting the following two points: (2, 5); (-3, 10)

Note that the difference between open and closed questions don't necessarily dictate the amount of time taken for each question.  You can have an elaborate closed question that consists of lots of words like the following example, but it would still only have one correct answer:

Bill wants to earn extra money selling lemonade in front of his house.  It costs $1.20 to start his business and each glass of lemonade costs $0.06 to make.  He plans to sell the lemonade for $0.10 a glass.
a) write an equation that represents his cost
b) write an equation that represents his revenue
c) graph both equations on the same set of axes
d) does bill make a profit or lose money for 20 glasses sold? 35 glasses sold? 50 glasses sold?

What makes open questions effective is the fact that students are allowed to demonstrate their understanding of their own volition.  On the one end of the spectrum, they provide opportunities to demonstrate in depth understanding.  Students need to understand so much more in order to be able to answer these questions. But because of the nature of the open questions, these questions naturally come with multiple entry points so students who may struggle with some concepts still have the ability to start answering the questions.  So it addresses the other end of the spectrum, since students who struggle with some major concepts still get the opportunity to demonstrate what they think they understand.

By the way, I am definitely not saying that closed questions are not useful -- they definitely still are.  I am just arguing for incorporating more open questions as an additional source of assessment of student achievement.

I highly recommend this book for those who are interested.  The book also recommends some strategies for turning your existing closed questions into open questions.  That is so much more effective than just having a wealth of questions (or "question bank:).  Afterall:

Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime
-Lao Tzu (old Chinese Proverb)

Small, M. and Lin, A. (2010) More Good Questions: Great ways to differentiate secondary mathematics.  (not going to bother citing it properly!)

Ok so let me summarize the main reasons behind my suggestions towards incorporating more open questions in our assessments:

1. Open questions allow for multiple opportunities for students to demonstrate their learning
2. Open questions allow for multiple entry points for students with different levels of understanding
3. Open questions arise more naturally and thus are inherently more fun (like these act 1 questions that students come up with themselves)
4. Open questions allow for student ownership of their own understanding and encourages them to organize, mediate, and combine the skills and concepts in order to present their answers.

I am sure there are more benefits of using open questions that I am missing -- but hopefully I've made a convincing argument for incorporating open questions.