Q: How can we as a country make the system-wide changes needed to help our students learn math more independently, by thinking on their own?
CS: The first thing it’s important to understand is that the U.S. is not consistent from school to school or district to district. In fact we have districts that are quite competitive internationally. The difference is that we don’t have that competitiveness on a national scale. When I taught overseas, other educators said to me, “American teachers are soft.” And it’s true, as Americans, we sometimes tend to be overly compassionate toward our students. We want everyone to succeed. We want everyone to do well. Most of all, we haven’t wanted to see them get frustrated because they can’t do something. That’s not necessarily the same philosophy I see in schools in other countries. I think we now realize that what we’ve done over time to protect our students from failure is actually lowering expectations, giving them less to do, and trying to explain everything we think they might need before giving them a challenging problem.
But today, what we’re doing more and more in this country—and perhaps we’re learning lessons from what other countries have done—is offering kids problems that might be more challenging and expecting them to do some struggling. We now know the value of what some folks call “productive struggling” or “constructive struggling”—when students really wrestle with a problem, and begin to think about things from a mathematical perspective, they develop an understanding that math isn’t always easy, but that they can figure things out. And they learn that what every person thinks about mathematics has value. It’s not just the teacher telling them the one way to do it, but in fact there may be multiple ways to do it.
There are many wonderful things happening in some classrooms across the country. But if the student’s next classroom isn’t structured around the same philosophy, they are likely to lose a lot of what they’ve learned. We need a stronger system-wide commitment—within schools, districts, states, and quite frankly within the nation, if we want to see the bang for our bucks. The truth is, if we want to see real results, then we need to have teachers working together toward consistent goals, certainly within the school, and preferably on a larger scale.
Accountability is important, but it doesn’t have to be high-stakes, multiple-choice tests at one time of the year. Really we ought to be keeping track of what kids are doing all year long using formative assessment strategies. And if teachers really learn how to analyze what their students are learning, I think we have a chance of expanding what we see in some classrooms, and schools, and districts across the country.
Q: How would you describe the approaches you recommend for teachers to help students learn to think mathematically?
CS: In terms of teaching strategies, we have what I call “upside-down learning” or what others might call problem-centered learning or student-focused learning. The trick is to turn the traditional teaching model upside down. As a young teacher, I was taught to explain things clearly (and do it with enthusiasm!), then do guided practice with students, and after that, give them some in-class problems and homework to try on their own. And what we now know is that students talking about their mathematics, thinking about their mathematics, and using that discourse as a basis for learning is a much more effective way to teach. That involves starting with a good problem they don’t already know how to solve. They have to think about it a little bit, struggle with it, talk about it in small groups and then as a class. Then the teacher skillfully identifies which are the most productive student conversations to pursue in depth. And it’s not always the ones that are right. Sometimes a wrong answer can lead to the most interesting mathematical discussion. A student may approach a problem from a misconception that other students may have as well, and in the conversation you may open up some possibilities for students to now understand the mathematics of the lesson.
So, essentially what we want to see is not necessarily every day looking exactly the same, but certainly having “rich tasks” for students to sink their teeth into, and using those tasks as a basis for student conversation—for student discourse—and the teacher facilitating that discourse: where are the mathematical ideas we can pull out, and how can we use those ideas as a springboard to the intended outcomes of the lesson? It’s not a matter of just letting kids mess around with math and hope they discover they’re supposed to learn. It’s a matter of using their thinking and engagement as a springboard to the mathematics of the lesson. Sometimes the intended mathematics comes out of those discussions directly and sometimes the teacher will build on those discussions to get to the mathematics.
So that’s how the teaching might look different. Professional learning might also look different. Teachers should be visiting each other’s classrooms, analyzing and planning lessons together. A lot of effective professional learning communities engage in analyzing students’ work with conversation like, ‘I’m noticing your students learned this better than mine. I’m wondering what you did differently in your classroom.’ Or even observing what others do in their classrooms. A more formalized version of this kind of professional collaboration is in-depth lesson study by a group of teachers working together.
Q: When we talk about constructive struggling and upside-down teaching, what are a few specific tactics you can offer teachers to get their students thinking about math on their own?
1) Start a lesson with a really good task that students can get into in different ways to start them thinking about the mathematics you want to engage them in.
2) Invite students to share their thinking, and structure ways for them to share with the class.
3) Listen to students and help them learn to listen to each other.
4) And learn to ask the kinds of questions that can draw out student thinking even further.
Q: How hard is it to learn to ask the right questions?
CS: Teachers often worry about what questions they should ask to get a student thinking. And it can be hard to come up with really good questions. But we can also fall back on a few general questions or prompts that we know work, like:
How do you know?
Why do you think so?
Say more about why you chose to approach it this way.
Show me where on the graph you can see the price of the [cellphone/shirt/…].
Explain how you got your answer.
What we want are open-ended questions or prompts that call for more than just simple, factual answers…questions that ask students to explain something, to share their thinking, even ponder a little bit.
Visiting other classrooms and hearing other teachers’ questions is also helpful. Write down what questions the teacher asks. This can be a great collaborative activity if teachers debrief the lesson together to talk about how effective various questions were for stimulating student thinking.
Q: How can districts help make this shift come about, and make it consistent throughout the country?
CS: That’s the challenge. Things are getting better. We’ve seen all kinds of improvements, particularly in elementary mathematics. I’ve seen widespread conceptual teaching that builds mathematical understanding with fluency—students know facts, skills, procedures and still have conceptual understanding. We even know some good ways to teach fractions, which is one of the most widely disliked parts of mathematics for both adults and students.
The real problem is how you scale up the “pockets of wonderfulness” that we have going on across the country, and that comes back to leadership and commitment, and making professional learning not optional. Professional learning needs to be more focused—not just choosing from a potpourri of options on a professional development day. School, district, and state leaders all need to say, ‘we are going to work together to give our students the advantage to build each year on what they learned the previous year so that they can extend their knowledge, their learning and their proficiency, year after year, consistently.’ To do that, you have to have teachers working together and participating in professional learning experiences together, and having professional learning communities that are really professional, really about learning, and really communities—not just an acronym (like ‘PLC’) that’s another name for a faculty meeting. Teachers need to have opportunities to dig into ideas and practice different ways of teaching and see each other doing it, and have conversations about what they’re doing.
Just as we want kids to be willing to share their thinking, we need teachers to share their thinking as well—to discuss how things work best in the classroom and how we can improve them.
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