I just finished working with a large group of teachers in Montpelier. They were inquisitive and eager. In the exit slips, several participants asked for more information about using questions to teach. This post is a response to these requests for more.
I tell everyone who will listen that the two most important elements of effective math instruction are:
The learner must do the work.
What the students have to say to each other is more important than what you have to say to them.
My goal in telling teachers this is to get them to have students make personal meaning of important concepts. It doesn’t really matter whether what you’re telling students makes sense to you. It only matters if they can make sense of what they’re doing. For this reason effective educators make questioning the most important teaching tool in their arsenal. It’s the “good question” that points students in the direction of discovering a key concept – “doing the work.”
I’ve seen good teachers find these questions (seemingly) effortlessly. When watching Sandi Stanhope (ALN Operations Manager and Facilitator) work with students on subtraction, for example, I have often heard her ask an important question that leads to deeper understanding. For example, in solving the expression:
Sandi will ask, “Where can I find 174 in 402?”
It’s a fabulous question. Not only does it point a student in the direction of taking 402 apart into 174 and something else. This question gets a student right into the essential part-part-whole nature of additive reasoning. With one question the student is drawn into thinking about an expression as both addition and subtraction without anyone telling them. This is the deep nature of “The learner does the work.” This good question presents the learner with an opportunity to construct important understanding on her own.
Good questions are powerful and sometimes subtle (more on that later). Sometimes, though, questions can be simple but the timing of them is critical. Anyone who has ever watched Karen Reinhardt (ALN Facilitator) do a number talk will know that her favorite question is:
“How do you know?”
Someone will say, for example,
“I knew four 25s make 100.”
Karen will ask, “How do you know?”
The nature of this question – and using it at that moment – asks the student to go deeper. Often students “just know things.” By asking, “How do you know?” after making, what appears to be an obvious, assertion, Karen is asking the student to think about the concept at a deeper level, to think about how he knows this is true. She is asking this because she knows that there is a potential connection between money and this statement. She wants the student to be conscious of this connection, too.
In both cases these skillful teachers are making use of their deep knowledge of the mathematics they’re teaching, and using that to make a little trail of crumbs to the important concept. They never tell students, “Here’s what you should know”, but rather, ask a question that will help students make their own leap to a new understanding.
So, what does one do when students don’t make a leap, even after a good question? Sometimes even the best question is not enough to lead a student to understanding. My advice is: Ask a smaller question.
Recently, for example, I was working with a student who was trying to learn to do some mental math. She was a relative stranger to doing math in her head, so she was having difficulty with most parts of the process. I was using the equation:
15 x 12
I like this equation because it makes use of 10s, 2s, and 5s – all of which are important for developing multiplication strategies. She looked at the equation and just shrugged.
“I don’t know. I can do it with paper.”
“Ok,” I said. “Do you know what 10 x 5 is?” This is a smaller question.
“50”, she said.
“What about 10 x 2?” I asked.
”Does that give you any ideas?” I was trying to leave her room to make a leap to decomposing, but she wasn’t able to make a connection.
“I don’t know what 10 x 12 is.” She said.
“How would that help you?” I asked.
“I could do 10 x 12 and then 5 x 12.”
I then asked her a series of smaller questions, trying to leave a trail of crumbs for her to follow conceptually. We wrote on some paper we were sharing.
“What’s 3x 10? 4 x 10? 8 x 10? 10 x 10? What do you notice about what happens when you multiply by 10?”
She looked at her answers on the paper and thought about it. Her face lit up and she said, “You add a zero at the end when you multiply by 10.” She was right… and she was wrong. I tried to head this off by reframing her statement:
“I think I see what you’re saying. You end up with a zero in the ones place when you multiply whole numbers by 10?”
“What?” She said.
“Are you actually adding anything? What happens when you add zero?”
She thought about this and then said, “Oh, right. OK. You’re putting a zero in the ones place when you make something 10 times bigger.”
I thought, OK, I can work with that.
Now I asked her, “So what’s 12 x 10?” She looked confused again so I helped her remember her own advice: “What did you tell me happens when you multiply by 10?”
“You put a zero in the one’s place because… wait, is it 120?”
“Yes!” I said. “Why?”
“Because when you multiply by 10, you put a zero in the ones place!”
This is an emerging understanding – I still want her to understand what happens multiplicatively when multiplying by 10 – but focusing on smaller questions allowed her access that she didn’t have when we began.
When a student is stuck, and I’m looking for smaller questions, I think about:
What is a concept the student would have to understand to do what she needs to do?
How could I ask a simpler question (or problem) to give her the chance to figure this out?
If I were thinking of the current problem as the end of a series of problems (a “goal” problem) what would previous versions look like?
Would using smaller numbers help?
Would using a simpler situation help?
Is there another piece of her work I could have her look at that might yield some insights.
As most ALN Interventionists can tell you, the first few attempts at smaller questions seem difficult. But, once you get the hang of them, the practice of asking a smaller question to encourage making a developmental leap will be second nature.