How do you know you are pushing your students’ thinking? The questions you ask in class can either push their thinking, and let them show you how they thought about solving a problem; or they can shy away if they feel they don’t have the right answers.
“Problem solving is the major theme of doing mathematics and ‘teaching students to think’ is of primary importance. ‘How to think’ is the theme that underlies genuine inquiry and problem solving in mathematics.” Erma Anderson 2019
Asking questions to our students, gives us the opportunity to see how they think, and gives them the opportunity to create a mathematical argument.
We know that effective teachers of mathematics ask purposeful questions. But what does that mean?
“Guess my thinking” type of question
This type of question is the one where the teacher has a specific answer in his/her head, and is hoping the student guesses it.
It could go like this:
T- “What is one strategy you could use to solve this problem efficiently?”
S- “A number line?”
T- “Yeah, it could be that but, what is one strategy you could use to solve this problem efficiently?”
T- “Mmmmm. Think about it. What is the one strategy you could use?”
S- “ I dunno.”
This conversation can go on, or it can stop right away. This question does not help the student figure out what strategy is more efficient. Also this type of question does not give the teacher any evidence of what the student thinking process is about, or what misconception might he/she have.
The most dangerous part of using this type of question, is that students shut down immediately, because they realize they are never going to guess what the teacher has in his/her head.
“How are you thinking about it?” type of question
This type of question allows the student to explain how he/she thought about a problem. It gives him/her the chance to pause and think about the steps he/she took, and what conclusion he has gotten to.
Here is where the teacher asks questions to find out what the student is thinking about (as opposed to a guessing game).
Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both the curriculum and instruction should begin with problems, dilemmas, and questions for students. – Hibert et al., 1996
Allow students to search for solutions, and resolve incongruities
A few weeks ago I was in a 1st grade class, working with a couple of kids working on the learning target: Represent and solve problems involving addition and subtraction. Ms. Jones had ask me to see if these 3 students needed some extension tasks, as they already were showing proficiency of adding 2 digits. We wanted to make sure they were not using only algorithms, and if they showed understanding of place value.
Ms. Jones and I did a number talk on using numbers flexibly to add and subtract. Then she gave them a few problems to solve with their partners. She gave them some paper with number lines on it, so they could use the decomposing strategy. For example when adding 26 + 25 perhaps they could start on the 26 on the number line, and then take a big jump of 10. Then 10 more, then 5.
After solving two or three problems, one student looked at me and said:
S- “Miss Caty, this is very silly. (He was giggling) I am not going to use the number line for this problem.”
Me- “Tell me more.”
S- “Well, look at the problem. If I am going to add 99 + 14, I don’t need a number line. All I have to do is take a 1 from the 14, and give it to the 99. Now I know I have 100 and 13.”
Me- “That is so true. Write that down.”
Allow students to wonder why things are
How do we know a number is even or odd?
Last week, our students in 2nd grade were exploring even and odd numbers, as part of their learning target: Work with equal groups of objects to gain foundations for multiplication.
One lesson gave them the opportunity to use color tiles, and make arrays with different numbers. They worked in pairs, and during the “discovery” time of the lesson, they came up with conclusions about why numbers 3, 5, 7, etc could not be arranged in arrays. One little tile was always left out.
Notice how the lesson was a DISCOVERY session, not a lesson to memorize what is even or odd.
Another lesson had students discover how even numbers could be split into 2 equal groups.
I was in 2 different classrooms that week, were teachers were working on the same lessons. We were very pleased to hear students make their own conclusions about why numbers were even or odd. In both classrooms, students needed the concrete manipulative to make sense of what even numbers look like.
However, in the next couple of days, when we sat down to look at student work, and analyze their “communicating & reasoning” in their notebook, we saw that several students were not using what they learnt in class to explain how they knew if the number was even or odd.
Look at this example here, where the student says that once you see the number in the ones place is 5, then you do not need to look at anything else!
Then he continues and explains that if you want to find out a number is even, then the number needs to have a partner. This second statement shows what he learned in class. The first statement is something he had memorized.
Another student, said that he knew a number is even when “apparently” you can count like this 2, 4, 6, 8. He also says a lot of other things, but the word apparently is on the 5th line. Take a look.
One student wrote that if you think about the word EVEN it has 4 letters, and if it is ODD it has 3 letters, so there you go. Think no more!
So we decided to ask more questions to see why they were using that rationale, after they had discovered in class how an even number can be split equally in 2 groups. Or also it could be even if every part of the number had a pair.
We finally realized that a lot of our students have memorized that “a number is even if it ends in 2, 4, 6, 8, 0” And a number is odd if “it ends in 1, 3, 5, 7, 9”
A lot of our students go to tutoring after school, and their tutors have them memorize all these concepts, without understanding why the rule works.
We asked questions like:
“What do you mean because 2, 4, 6, 8?”
“What makes you think 5 is odd?”
“What did you discover when you use the color tiles to make arrays with numbers 0 to 20?”
The good thing about asking purposeful questions, is that if you ask to find out what they are thinking about, or why they says what they say, you can find out the misconception, and with a few more questions, students can make connections between those thoughts and their learning experiences. You will hear a lot of “ahhhh” and “ohhhh”, when students make connections, or when they finally understand why that rule they memorized, makes sense. Memorizing in not fluency.
We want our students to be fluent in mathematics. “To be fluent means not just being able to do something, or memorize something, but being able to reliably do it accurately and efficiently. Fluency is an outcome of a progression of learning and sufficient thoughtful practice provided at each grade.” Erma Anderson, 2019
Do you want to find out what your students know about even and odd?
Use “How are you thinking about it?” type of questions, to find out!
These are some samples of student work, after we conferred with each of those students, and they had one more chance to show what they learned, not what they had previously memorized.