# To B or not to B

What do you want for dinner?  Pizza or Hamburgers?  That’s like asking would you rather have 2x \$10, or 4x \$5.  Some questions don’t have a clear answer, but our students for some reason seem to think that everything has one clear answer.

Little kids play the game all the time, “Which of these is not like the other?”  Being taught to analyze irregularity and develop pattern recognition is central to decision making.  What is the most common decision students have to make?  A,B,C or D (and sometimes E).

For years this format has been consistent, and simple.  Choose the correct answer (or at least take your best guess).  One would hardly suspect that more than one response could be correct, especially at the same time.  Imagine the following prompt:

Which of the following is a fruit?

• Apple
• Banana
• Cherry
• Guerrilla

Which answer is correct?  Should I assume the prompt had a typo?  For example should it have said, “Which of the following is not a fruit?” or “Which of the following eats fruit?”  The current culture is that having more than one correct answer is more of a paradox than a reality.

Even more paradoxical is the presence of such questions in standardized testing.  The United States is anticipating multiple response questions (having more than one correct answer) from institutions like SBAC and PARCC.  I can imagine the varying depths of knowledge in choosing both correct answers, only 1, or some type of combination in between.  Let’s say we assign 1 positive point for each correct answer chosen, and -1 point for each incorrect answer, and a zero value for each choice not selected.  This would give a perfect score of 3, and a minimum of -1.  Consider the following outcomes:

 Apple +! Banana +0 Cherry +0 Guerrilla +0 +1 Apple +0 Banana +1 Cherry +0 Guerrilla +0 +1 Apple +0 Banana +0 Cherry +1 Guerrilla +0 +1 Apple +0 Banana +0 Cherry +0 Guerrilla -1 -1 Apple +1 Banana +1 Cherry +0 Guerrilla +0 +2 Apple +0 Banana +1 Cherry +1 Guerrilla +0 +2 Apple +1 Banana +0 Cherry +1 Guerrilla +0 +2 Apple +1 Banana +1 Cherry +1 Guerrilla -1 +2 Apple +1 Banana +1 Cherry +1 Guerrilla +0 +3 Apple +1 Banana +1 Cherry +0 Guerrilla -1 +1 Apple +1 Banana +0 Cherry +1 Guerrilla -1 +1 Apple +0 Banana +1 Cherry +1 Guerrilla -1 +1 Apple +1 Banana +0 Cherry +0 Guerrilla -1 +0 Apple +0 Banana +1 Cherry +0 Guerrilla -1 +0 Apple +0 Banana +0 Cherry +1 Guerrilla -1 +0 Apple +0 Banana +0 Cherry +0 Guerrilla +0 +0

Do these seem fair to you?  What if there were only two correct answers and Apple was some different response like Animal?  This changes the range to anywhere from -2 up to 2.  Using the same method of calculation , +1 for correct and -1 for incorrect, we would be valuing each multiple response question according to the total number of correct responses possible.  Is having 2 correct answers less valuable than 3?

Recently I was told a story about how some of the new institutions plan to handle scoring these new multiple response items.  Because of some disagreement and outcry of the original plan, the new policy would be a total score of +1 for choosing all correct responses, and +0 for anything else.  In other words, there is no partial credit, or stepped out scoring.

I’m a fan of having questions that slow down thinking, and make the reader consider more possibilities.  I think having multiple responses correct is a good thing.  It encourages argument, which requires reason.  Isn’t that why we practice math in the first place; to apply reasoning.  Especially when two choices seem to be both right.

How do we help kids answer the paradox one feels in having more than one correct answer?  Is there a fair way of programming feedback for all the possibilities?  Should we even try, or just go back to good ol’ choose A or B (or C, or D)?