Always, Sometimes, Never Shapes

I know ASN has been around for a while, but I am still amazed at how such a small twist on asking students questions to determine knowledge will bring out such great thinking and understanding.  I remember going through class and having these two questions appear on a test:


___ A rhombus is a square.

___ A square is a rhombus.

These types of questions were landmark at the time I’m sure, but when I think about how much knowledge I displayed by answering them- it’s very disappointing.  It’s a T/F question, which means even if I don’t have any clue what it is asking I have a 50% chance of getting it correct! (And who didn’t like having T/F Q’s on their tests?)  I write T or F, and there is no justification of how I know either statement is true.  Even if I was asked if it’s true my response would involve: “A square is always a rhombus”- a simple restatement of the original question.  My math teachers would either acknowledge my response or tell me I was wrong and why.  After being wrong enough, I finally figured out what I needed to say to get the answer right- not that I understood the distinction.  Now I ask students a much different type of question:

Always/Sometimes/Never (explain your response- how do you know?):

______ A rhombus could also be called a square.

The biggest shift in this question is that it is no longer presented as an absolute fact- something that students had to memorize.  Students have three choices, which means they need to consider all the possibilities of answers and the statement as well.  What does a rhombus look like, what it is properties, how many different possibilities of rhombuses are there?  They then go through the same process for squares.

Here are some responses by my students:

  1. Sometimes: because it is kind of shaped like a square.  People could refer to it as a square but other people may not.
  2. Sometimes because a square is classified as a shape with four equal sides.  If a rhombus has equal sides it is a square
  3. Sometimes because it’s a slanted diamond square
  4. Always because a rhombus is the same as a slanted square.

Now, I threw this out there to my students because I am going to start a geometry section and I wanted to get an idea of their understanding of shapes and properties.  Many students said that they didn’t remember anything about shapes and properties, and their justification of thinking supported that claim.  This question did give me a great idea on student’s background knowledge on it however, and I have a good idea of how I want my unit to begin.

These are other questions I plan on asking….


I can’t wait to hear student thinking and the discussions that will occur from these.  Many of my students are afraid of Geometry- I have one who admitted he skipped class every day.  He hated proofs.  We talked about that as a class- and how more traditional teaching could lend a hand to their fears.  We also talked about how ASN went, and how we explained our reasoning and justifications.  I then told them that they just did a mini-proof.  Being comfortable with explaining why you do things in mathematics supports proof reasoning.  I also told them that I would show them how easy it is to transition from the group discussion we just had into proofs.  They relaxed and were good with that, I can’t wait to “see” their thinking over the next unit.

Let me know if you have any other favorite ASN geometry Q’s- my students enjoy hearing from you.

2 thoughts on “Always, Sometimes, Never Shapes

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