A Maths Teacher who uses Open Questions

The students had found answers to the square root of target numbers such as 81, 100, 9000 and similar numbers. They had also used calculators to look up the square root of 80, 1000, 9.5 and similar integers and decimals. So far, so good.

Now they were looking for the square root of -81, -100 and -9000. Their calculators were displaying the word ERROR. So were the looks on their faces.

The teacher wanted the students minds to be fully open and ready fior what was to come next. For many students, this topic is a watershed moment. Success could move them towards being mathematicians; failure could stop them on their journey.In essence the conversations with the students, one-to-one and in small groups, went like this. Notice that most of the teacher's questions start with "What...", some are closed questions, others are leading...

T = Teacher
S = Student
​
T What’s the problem?
S (Looking at their calculator they said) It says ‘ERROR’
T So what’s the problem?
S We can’t answer the question.
T What are you looking for?
S A number that gives us -81 when multiplied by itself.
T How will you know when you have the right answer?
S When it doesn’t say “ERROR”
T What’s special about ‘square root’?
S It’s what you need to answer the question.
T You mean ‘the number that when multiplied by itself will give the target number’?
S Yeah
T What’s the target number?
S Eighty-one
T Is it?
S No, it’s minus eight-one.
T And what have you tried?
S Nine
T And what have you not tried yet?
S Ah. Minus nine
T And what do you get when you try minus nine multiplied by minus nine.
S Eighty-one
T And...
S Is that it then?
T What’s the target number?
S Eighty-one
T Is it?
S No, it’s minus eight-one.
S So minus nine isn't the answer
T No
S So what is? Can't you tell us?
T Not yet. What else have you tried?
S Nine, and minus nine
T And what have you not tried yet?
S Decimals!
T And what do you get when you try a decimal?
S Like nine point zero?
T Give it a try. What do you get?
S Eighty-one. But not minus eighty-one.
T So what have you not tried yet?
S Can't you just tell us?
T Yes, but that would be stealing the learning from you!
S We have tried everything!!!
T You have tried whole numbers and real decimals. So what kind of number do you need?
S We need something else!
T Are you ready to hear about a different kind of number?
S There's a DIFFERENT kind of number???
T Are you ready to hear about a different kind of number?
S Yes.
T Then I am ready to tell you about …