……… … Silence …
27 faces looking at us. Expectantly. Politely. Silently.
The two of us (Kathleen and Corrinne) have visited dozens of classrooms over the last few years, on a mission to help Kg-6th grade teachers in their quest to improve math education in the US. The first time we failed was in a 4th grade classroom, 8 years ago, where we’d posed this question:
We had expected the students to huddle together – talking, arguing, giggling – until the problem was solved. We thought they’d use guess-and-check, or push around little bits of paper with 3’s and 7’s written on them. We’ve been teaching and mentoring teachers in our own school for almost 20 years, and have grown used to the engrossing messiness of problem solving.
Instead … silence.
Not much else went well that day. So a couple weeks later, we tried it in a different 4th grade classroom. This time, we provided the slips of paper with 3’s and 7’s on them, thinking that would scaffold the problem-solving attitude we wanted.
Again, not much happened. We were baffled; what was going on here?
Finally, one girl raised her hand. “Ma’am?” she ventured. “Umm, No one ever taught us that problem before.”
Ahh. Cringe. Memories came flooding back… Introducing a problem-based curriculum at our own school 17 years ago was grueling. We had to do battle with the established culture of “How Math Is Taught” – namely by explaining, practicing and drill. When we challenged that paradigm, we met with student silence, parent resistance, administrator concern, and our own self-doubt.
Thank goodness there were two of us. Alone, either one would have given up.
What makes this so hard? We teachers mean well. We want to teach. Something in us yearns to take the class by the hand, and say “Make an organized, 2-column list. Write ‘3¢ stamps’ as a heading on the left side, and write 1, 2, 3, 4… below that. Now label the right-hand column ‘7¢ stamps’, and in each place, put the number that would make the pair of numbers add up to 20….”
Our question to ourselves, and to teachers we work with is:
“Who is doing the thinking here?”
We’ve all read the research. Citizens of the 21st Century will need thinking skills, creativity and brainstorming habits much more than they’ll need algorithms and computational speed.
We truly believe this – Nowhere can we teach thinking skills better than in the math classroom.
So – Nuts and Bolts: How do we begin the change to “Building a Thinking Classroom”?
- Ask, Don’t Tell
Answer student questions with questions! Yes, this is hard, so here’s a countdown of our favorite responses:
#5. “Hmmm… What do you see so far?” (note the verb!) Just get them talking. “Humph. I see two colors of blocks” is an okay answer to start with. “Oh, sure. Which color has more? How many more?” Anything you ask is okay, too. Moving from the concrete to the abstract is hard, and the bridge between the two levels is talking. Just get them talking; progress will follow.
#4. “Can you see a pattern? Is it growing or shrinking? How many are in each group? Can you count them?”
#3. “What would happen if….? … we used simpler numbers? we acted it out? We took away the extra numbers for a minute? We lumped both numbers together for a minute?” These are not just fillers – these are responses that actually help move students forward.
And our #2, so close to our hearts:
#2. “Have you asked the blocks?” (We call base-10 blocks the “math whisperers”)
* * FAQ: But… what if students answer “I don’t know, I don’t get it” to every prompt above? **
Then we go on to our favorite response :
#1. “I don’t know…. Let me try it with the blocks.”
Yes, play dumb. By “not knowing” and getting the blocks, you’re modeling problem-solving behavior. It might raise their hackles the first time, but by week 2 or 3, they’re hooked. Here’s the thing – you’re giving students respect for their inherent intelligence. (This is very powerful – think about a time you didn’t feel your intelligence respected.) Here’s our take on ‘thinking’:
We want to pay tribute here to another educator whose writing has been invaluable to our own growth: Peter Liljedahl, whose website can be found here, and whose phrase “the Thinking Classroom” we’ve borrowed.
2. Encourage Counting
If students learn to rely on blocks, manipulatives and drawings, it means they’re functioning at the visual level initially. It means the answer is usually right in front of them, on the table – in the blocks, but they’ll have to count to distill an answer from that visual.
There’s nothing wrong with that! Real math evolved from counting. Any major concept you really understand can be reduced to counting. Slope in algebra? Count. Sine and Cosine? Count the degrees around a unit circle of 2 pi. Find a derivitive in calculus? Count the slope of a tangent to a curve. (isn’t YouTube great?). Our point is that there is nothing wrong with counting – math is just an abstract set of symbols developed so we don’t have to count – just to save time. Let them count. The human brain loves efficiency – they’ll figure out how the math just to avoid counting so much.
The one response we do make them memorize (!) has to do with drawing word problem models. “Class, why do we draw models for these problems?” Their required response: “So we can count”.
3. Give Judicious Hints
Even in a Thinking Classroom, students get stuck. After letting the class work for a while, we watch for signs that the task is becoming frustrating, not fun any more.
Then we call the class together and say “I wanted to share a great suggestion from one group. They said (even if it’s not true ! ) ‘What if we pretend all 20 of the stamps are 3¢ ? Could we then start switching out stamps?’ ” It doesn’t seem like much of a hint (“Try using twelve 3¢ stamps” would be a thinking-stopper). But it’s enough to get the students around 3rd base.
4. Accept All Answers
If students ask “Is this answer right?”, we say “I don’t know – I’m still working on it.” But we write up ALL the answers that get suggested on the board (and sometimes add a few wrong ones of our own, so the correct answer doesn’t dominate). This encourages students to look at each others’ answers and check their own work.