“In mathematics the art of proposing a question must be held in higher value than solving it.” (Georg Cantor)
The art of proposing a question is critical in our math classes because as educators we want to promote thinking and reasoning. Some questions posed by adults can be very closed and may only require memorization of a formula or a procedure. Once students engage in solving questions, the art of questioning on the part of the educator or parent can either move student thinking forward or shut it down. As a teacher I am always aware that it is the student thinking I want to make visible because it is through the discourse that misconceptions come to light. So while students engage in the solving of the problem my focus is not on the correct answer but instead the thinking made visible through my questioning (e.g., effective use of talk moves as described by Lucy West).
In addition, I think that we need to listen carefully to the questions that students ask and view those questions as a vehicle into their understanding and an opportunity for their own inquiries. Sometimes as educators and parents we feel that we must answer all their questions when really what we should be doing is turning it back to them and saying, “Mmmm I wonder why that is. How can we find out!’
This past week I was in a Grade 5-6 classroom. The lesson was on place value and creating 5 digit numbers. The teacher began the lesson by reviewing the learning goals being explored; writing numbers in expanded form, representing numbers in different ways, and decomposing numbers. The teacher and I discussed that the big idea behind this is that we want the students to understand that large numbers can be decomposed into thousands, hundreds, tens and ones in many different ways. Can students who are able to write a number in standard form understand that the number can be written in expanded form in different ways and still maintain the same quantity e.g., 500+40+2 or 400+120+2 or 500+30+12 (concept of regrouping). Place value can be taught in a very procedural and rule oriented way. This ability to decompose numbers in several different ways impacts students’ fluency and flexibility with numbers. One of the first questions asked on the task sheet was about creating a 5 digit number with an 8 in the thousands place. Almost immediately a student asked: How can you make a 5 digit number with 8 thousands? After the lesson the teacher and I discussed how this student question gave us insight into a misconception or a fragility with place value and that I view student questions as data and an opportunity for inquiry. I might respond to this student by saying that Joe has just asked a great question. I wonder if that is possible? Does anyone have any thoughts about that? or Let’s find out if it is possible. After the lesson I shared the following observations with the classroom teacher.
Some Student Observations
Joe: Question: How can you make a 5 digit number with 8 thousands?
Barry had 40 for 4 in the tens place as asked but had a 2 digit number only. I pointed out that he needed a 5 digit number. He promptly changed it to 40 000. I asked if the 4 was still in the tens place. Barry said no and changed the number to 60 400.
Sarah: 90 876 – she pointed to the 9 and asked if this was 90 000 or 9000. I referred her to another number she had created. She was fine with calling the 4 in 43 624 40 thousand but the 0 in the thousands place confused her in 90 876.
James: when creating 5 digit numbers was able to put a 7 in the hundreds place but was challenged to put a 9 in the thousands place.
My Question or Wondering: What is it about 5 digit numbers that makes going to the thousands and ten thousands challenging? Is it because 92 000 has a 2 in the thousands place but we say 92 thousand. I’m thinking that these students who can recite that there are no tens in the tens place in 100 may be challenged to understand that there are indeed 10 tens in 100. This is why the learning goal related to decomposing numbers in different ways is so critical to their understanding of place value and quantity relationships.
After analyzing the independent piece (exit cards) at the end of the lesson the teacher noted patterns and challenges that informed his next day’s lesson. At one time he may have moved on but now he is slowing down based on the students in front of him and going deeper knowing that the time spent on developing his students’ fluency and flexibility with big numbers will pay off down the road when he is working on other math concepts.
Jo Boaler, a professor at Stanford University says that math is full of uncertainties and that mathematics should be about posing good questions and then exploring them. In this classroom, the teachers (classroom and coach) and students are doing this and as a result learning from each other!