Recently, Desmos has been building out its platform to offer customizable lessons. Led by Chief Academic Officer Dan Meyer, a former math teacher who left the classroom to pursue a PhD in math education, Desmos has been using its platform to model how technology could change pedagogy.

Desmos tries to harness the social nature of online interactions into meaningful math inquiry. Meyer says students love the internet because it’s a social place to share and create. And, a math classroom at its best is also a place where students are creating hypotheses, testing their thinking, critiquing each other’s work and discussing how and why mathematical laws work.

“Typically, online math platforms have no concept of the student in relationship to other students,” Meyer said in reference to “personalized” programs where students work through a set of problems or concepts “at their own pace,” but do so in a vacuum. Meyer argues this model doesn’t capture what’s powerful about a class full of students.

In contrast, Desmos allows teachers to make a series of slides with interactive elements. One slide might have a video of a glass filling with water, with a question asking students to graph it. When a student submits his graph, a Desmos default function then shows the student three other student answers and asks him to give feedback on the solutions. Teachers can shut off this function, but Desmos intentionally made it a default to encourage discussion.

“Our core assumption is that students need to be aware that there are other students in the class, and that refining process is part of every math lesson,” Meyer said. Teachers could also have students create their own glass filling at a specific rate and ask them to graph that. Or, maybe each student submits the problem they devised for other students to work on. Teachers can also display all the solutions, with or without student names, and ask the class to analyze each other’s strategies.

Meyer hopes to support teachers as they design lessons that have some common elements of good math instruction: Students are thinking beyond equations, the learning is social, processes are made visible and students get written feedback, often from peers. “Getting numerical or binary right-wrong feedback tends to make the student think about the self,” Meyer said. “Written feedback about the work tends to focus on the work itself.”

**HOW ARE TEACHERS USING DESMOS?**

Middle school math teacher Cathy Yenca started using Desmos when her district discovered the graphing calculator tool was going to be integrated on state tests in Texas, where she teaches. Wanting her students to be familiar with the tool, she started experimenting with it in the classroom and it has now become an instructional “necessity.” Yenca works at Hill Country Middle School, a public school in Austin. Her students all have iPads and Yenca is passionate about the power of tech in learning, but she hasn’t liked a lot of what’s out there for math specifically.

“When you come across something that’s not just skill, drill, kill, and is kinda rich, it gets your attention,” she said. She uses Desmos because it makes inquiry in math class easy.

For example, in one recent lesson with her algebra one students, Yenca was teaching transformations in quadratic functions. But she didn’t tell students that’s what they were studying. Instead, she set up a task where through exploration they were telling her the lesson by the end of class.

“Watching how they change a parameter and that instant feedback of what they just changed and how that impacts a graph, they’re hooked,” Yenca said. “When you have that reaction from middle schoolers around math, that’s a win.” She sees how peers influence one another’s thinking, and that even if not every student is on the right track the whole time, they are figuring it out together.

Yenca often takes a moment to show the whole class everyone else’s answers. This overlay would typically be what a teacher would use for formative assessment, to check for understanding, but Yenca said, “to me, keeping that for the teacher’s eyes only is a disservice.” She says learning this way requires building a classroom culture that values mistake making, but once that’s in place, so much rich learning comes out of students being able to see the trends in misperceptions. Together they discuss and untangle thinking until they’ve arrived at the math concept.

In a lesson with her eighth-graders on shapes in the coordinate plan, Yenca had students creating reflections and dilations on a Desmos graph. “Desmos is only going to do what you tell it to do, so if it does something you didn’t expect, you’ve got to figure it out,” Yenca said. Her students manipulate the variables and gradually come to an understanding.

Yenca says some of her colleagues think Desmos does too much for students, making graphing too easy. She says if rote graphing is the goal, then yes, Desmos does too much, but wonders if that’s the right goal. “If we are concerned that a graphing tool can graph for our kids, maybe we need to ask more of our kids,” Yenca said.

While Desmos is trying to make it easier for math teachers to incorporate these elements into classrooms, the platform doesn’t force the issue. The tool is completely open; teachers shape their own lessons within it, and could easily make something that looks essentially like a worksheet.

**REDEFINING CLASS**

Audrey McLaren teaches at a virtual school in Quebec. Her classes are all completely online, although they happen in real time, with students participating as they would in a brick-and-mortar classroom. Most of McLaren’s students live in rural places and their local schools don’t offer the courses she’s teaching in English.

“We’re not in the same physical space as our students,” McLaren said, “so we couldn’t see what they were doing until Desmos activity builder came along.” Now, she can pose a problem to students and then watch as each student tries to solve it. She can look at the class as a whole or zoom in and interact one-on-one with a student. McLaren thinks students are participating more than they would be in a normal class because every student has to do the work and share their thinking, whereas in a typical classroom (where she taught for 20 years) only about 10 percent of students raise their hands and participate in discussions.

“I try to design things so that after three to four slides, or questions, I stop everybody and within the online environment we put everything up on the board, classify the findings, and talk about which ones they agree with and why,” McLaren said. She uses the early slides to let students have a discovery period, where they’re playing with a concept, developing hypotheses and looking for patterns. “I want them to get an intuitive sense of what I want them to know; I don’t want to just tell them,” McLaren said.

She appreciates how Desmos will put forward a new kind of activity, let teachers play with it, and then open the tool so educators can build something similar. For example, Desmos has a few math “games,” but unlike many games that are basically practice with a prize at the end, the Desmos games make math knowledge central to completing the task.

In a Polygraph, for example, a pair of students might be given 16 graphs that all look different, but are all linear functions. One student chooses a graph and her partner has to guess which graph has been chosen by asking “yes” or “no” questions, a bit like the game Battleship. Students have to use math vocabulary and knowledge of terms like slope and y-intercepts to eliminate various graphs and zero-in on the correct choice. “You learn math from playing the game itself,” McLaren said.

At first Desmos created these Polygraph lessons around different common curricular topics, but now they’ve opened it up so teachers can make their own. She often uses Desmos in class as a way to explore a concept and then has her students watch a video at home to nail down the concept. But she also uses Desmos to deepen understanding in the middle of a lesson and as formative assessment at the end as well. Since she’s been using the Desmos activity builder for only a few months, McLaren doesn’t have any data to prove that teaching this way is improving math achievement. But anecdotally she’s confident it has increased participation, which should increase understanding, and she’s been impressed at how her students are discussing and writing about math with one another.

**TEACHER COMMUNITY**

In Bob Lochel’s Advanced Placement statistics classroom, getting technology to each student is a challenge. Even though he teaches in an affluent suburb of Philadelphia, his students don’t have one-to-one access and booking a computer lab can be a pain. So often Lochel relies on his students’ personal devices for access to Desmos. He’ll ask students to complete a few questions and then, like McLaren and Yenca, he often projects multiple student answers on the board as a jumping-off point for a discussion. Student are critiquing one another’s thinking. “That’s not the kind of thing we were asking before,” Lochel said.

One of his favorite parts of using Desmos is the community of math teachers that comes with it. Every educator spoke about the collaborative community of teachers sharing ideas using the hashtag #MTBoS. Many of the active educators in this community also write their own blogs, where they track the success and challenges of different lessons. Teachers can upload their lessons to Desmos as well, making it easy to find and use all or part of another teacher’s work for their own purposes. Lochel said often if he’s putting together a lesson and isn’t quite sure if it’s reaching the mark, he’ll put it out to the community for feedback.

Just like developing all good lessons, Lochel said it can be tricky to design a Desmos activity that both allows students to be creative and inspired, but also drives towards the ultimate goal for the class period. He appreciates the virtual community of educators that are helping him refine this skill. Lochel said when a lesson successfully allows students to arrive at their own conclusions, like the one he did on binomial distribution and how it’s linked to normal distribution, students understand in a much deeper way. Instead of telling them the rule, “this time they discovered the rule,” he said. And the buy-in that creating the rule engendered meant that they could also debunk the rule.

Desmos employees like Meyer, for their part, are constantly working with teachers to improve what the platform offers, while balancing a desire to seed good teaching practices. Meyer said while thousands of teacher lessons have been uploaded to Desmos, only a fraction are available through the search tool. Those are the lessons that he and his educator team have hand-polished, reaching out to the original author for permission, and re-releasing. He also looks at a random sample of teacher-created lessons every week and believes the quality has gone up over time.

The best part of Desmos for many educators, whether they are using it only for its graphing calculator capabilities or for these more involved, inquiry-based lessons, is that it’s free. That’s possible because Desmos licenses its calculator tool to curriculum and testing companies. The fees from that work fund the curriculum development and training work that Meyer does. He’s hopeful that before too long he and his team, in cooperation with teachers around the globe, will have developed what amounts to an Algebra I curriculum designed entirely out of low-floor, high-ceiling Web-based tasks like the ones described in this article.

“I was worried that we couldn’t figure out how to make “good” work in the market, but it’s been nice that we’ve found traction with paying customers,” Meyer said. “Part of that is the product and part of that is that there’s been a sea change in online math education.”

]]>Perez says when students are engaged, predicting answers, talking with one another and sharing with the class in ways that follow safe routines and practices, they not only achieve more but they also act out less. And everyone, including the teacher, has more fun.

“If we don’t have their attention, what’s the point?” Perez asked an audience at a Learning and the Brain conference on mindsets.

She’s a big proponent of brain breaks and getting kids moving around frequently during the day. She reminded educators that most kids’ attention spans are about as long in minutes as their age. So a third-grader can concentrate for about eight minutes before losing interest. It’s a teacher’s job to make sure there are lots of quick, effective brain breaks built into the lesson to give children a moment to recalibrate. Perez says teachers must be prepared for a diverse cross section of learners with a large toolkit of strategies for teaching in multiple modalities, with many entry points to participation and content.

**PEREZ’ BRAIN-BASED STRATEGIES**

**1. Don’t Be Boring
**“In our engaging classrooms, we have to have a set of procedures and routines,” Perez said. But they don’t have to be boring. She often has students come in and look at a list of adjectives on the board, many of which stretch her students’ vocabularies. She asks them to greet two other students and use one of the adjectives to describe how they are feeling today. The activity gets them up, moving and ready to learn, plus they’ve used a new vocabulary word in relation to themselves, checking in with their community along the way.

**2. Vote**

Activate students’ brains with a quick round of voting. Perez often puts three learning goals for the day up on the board and asks students to vote for the one they think is most important. All three goals are good ones and there’s no wrong answer. “The reluctant learners get to look around the room and see who else thinks just like them,” Perez said. This quick activity helps create curiosity among students about what each of them is thinking.

**3. Set Goals**

Perez is also a proponent of both teacher and students setting personal learning goals every day that are achievable, believable and measurable. “Part of reaching that goal is publicizing that goal,” Perez said. Making goal-setting a regular and visible part of one’s teaching practice models it for students. But it’s very important to leave time for students to revisit the goal they set at the end of the day, Perez said. That opportunity to reflect will help them see and value what they did during the day, as well as where they may have fallen short of the goal.

**4. Form Groups**

Perez constantly asks her students (in this case a group of educators) to break off to share with one another, brainstorm or collaborate, and she always sets a time limit for the conversation, like 72 seconds. “In my classroom I use bizarre time limits and then they think I’m actually watching the clock and they get to it,” Perez said. She finds this promotes more time on task than a generic five-minute time limit, which students know is just as likely to stretch into eight minutes.

**5. Quick Writes**

Often Perez will throw out a question and ask students to quickly brainstorm on paper as many answers as they can. Then she’ll do a “popcorn share” where students stand up whenever they want and throw out an idea. This could be an alternative to something like “round-robin reading,” which can put reluctant learners in the hot seat. In this case, Perez sets her students up for success by giving them time to brainstorm first — the answers are right in front of them. This strategy has the added value of forcing students to listen closely to their peers, since they don’t know who will pop up next.

**6. Focus on the ABCs: Acceptance, Belonging and Community
**“Without this set of ABCs, traditional ABCs will not be as successful,” Perez said. She’s aware of the rush to cover content in many schools and classrooms, but says teaching is not about what is covered today, it’s about what is uncovered in students. “Don’t be so standards-driven that you forget the needs of your students,” she said.

**7. Continually change the “state” of the classroom
**These are changes in who is providing the information, who is doing the talking. Perez likes to say for every 10 minutes of content, teachers need to give students two minutes of “chew time.”

**8. Empathize**

Keep in mind the students’ perspective and listen when they explain what they need to learn. Take Ned’s Great Eight to heart.

**NED’s GREAT EIGHT**

- I feel OK
- It matters
- It’s active
- It stretches me
- I have a coach
- I have to use it
- I think back on it
- I plan my next steps

**9. Do a BRAIN checklist**

- Build a safe environment
- Recognize diversity in the classroom
- Assessment must be formative, authentic and ongoing
- Instructional strategies should be a palette of opportunities
- New models

“We’ve got to be growing and open to new ideas,” Perez said. “That’s why teaching is such an adventure. Each day you walk into the classroom, you never know what you’re going to get.”

**10. Simplify**

Perez suggests framing every lesson in a similar format, but executing it differently each time. First activate the learners by making them curious and developing a need-to-know. Then, let them dig into the content in an exploratory phase that takes them deeply into rich content. Last, help scaffold students’ broader understanding by helping them integrate it with what they already know. Some metacognitive questions that can get them thinking this way include: What part of the lesson did you like the best? What part was the most difficult for you? Why do you think that was? What do you think you can do today to help yourself stay focused?

“If we don’t give our kids time to reflect, to connect, to marinate on the information, they’re going to regurgitate what’s right there in front of them without even thinking,” Perez said. Reflection and rehearsal of what was learned is crucial to move information from working memory into long-term memory.

**11. Chunk Information **

Make information more easily digestible for students. “We need to be more purposeful in our delivery of information,” Perez said. Too often teachers deliver an entire lesson without letting students move or discuss once. Kids will give up if they are overloaded with facts, and chunking provides a way to pause and let students think over what they’ve learned. Breaks to assimilate information are crucial for mastery.

“Lesson mastery means students have mastered the content when they do something substantive with the content beyond echoing it,” Perez said.

**12. Props**

Perez keeps a box of props for when she’s teaching. She often throws something to a child when it’s his turn to talk so he has something to focus on. She says this works particularly well for kids with attention problems, as well as for the tactile learners.

**13. Breaks**

Short video clips can be a great brain break. A great clip can be interpreted in multiple ways. “You’re fostering divergent thinking,” Perez said.

**14. Post-Its**

Post-It note discussions are a good way to get all students involved without making anyone uncomfortable by putting them on the spot. Ask an open-ended question. It could be an activator at the beginning, a marinator in the middle, or even a summarizer to test for understanding at the end of a lesson. Students jot down their answers to the prompt on Post-Its. English Language learners or special needs students could write just one word or draw something. Then students share in pairs. “Even the most reticent learner is OK sharing one-on-one.” Perez said. Post all the responses on a graffiti board and pull out some trends.

**15. Make Snowballs**

The Snowball brain break is one of Perez’ favorite ways to summarize learning at the end of a lesson (and should be done when students are on their way to recess or at the end of the day). Students write answers to a prompt on a piece of paper. On the count of three, they throw their “snowball” randomly up and away (but not at anyone). Then everyone grabs a snowball that landed near them.“It’s a way you can purposefully pause, have them reflect and make connections,” Perez said. She uses it in all subjects, sometimes asking students to write three new vocabulary words they learned, or three successes they had in that lesson, or three questions. “Students love it and it’s inclusionary because it’s anonymous,” Perez said. Students also get to see one another’s thinking in this activity.

**16. Guessing Games**

When slightly boring content must be covered, create a need-to-know in students by having them predict the answers. Students are more likely to be invested in the answers when they are revealed after students themselves have had a chance to debate and predict.*

This strategy among others is meant to get students to manipulate and think about the information themselves. “If the teacher does all the interacting with the material, the teacher’s brain, not the students’ brains, will grow,” Perez said. That’s why Perez advocates that teachers have a large toolkit of approaches to get students thinking, speaking, writing, touching, building, listening and, most importantly, doing something with the content.

**17. Balanced Inquiry**

Lectures do have a time and a place, but they are far more effective when they are interactive. Perez likes Harvey Silver’s guide for an effective lecture: connect new knowledge to existing knowledge, organize the materials into chunks, dual code the information so it’s stored in multiple places and exercise the brain.

“It’s a matter of balance to keep the engagement alive,” Perez said. She doesn’t advocate that teachers always have students teach one another just because it has a high retention and transfer rate; doing all of one thing is never effective. Instead, she says, it’s about a balanced use of all the inquiry approaches.

**18.** **Mind-streaming**

Mind-streaming is another fun brain break activity that also gives students a chance to recall what they’ve learned and teach one another. Have students randomly pair up and then each person teaches the other the most important things they’ve learned in that lesson. Each person will remember different things, and when there is overlap that will reinforce the concept. It’s simple, effective and doesn’t require any teacher preparation because students are teaching one another.

**19. Be Interactive**

Perez begs educators to always try to make tasks engaging and interactive by giving students enough knowledge, giving them the language to express it, giving them an authentic reason for the interaction they’re engaged in, prime them with interesting questions, establish a community of learners that support each other, and give students a clear understanding of the task. If these elements are part of every class, she says, all students can be successful.

**20. HOPE**

The last tip Perez offered educators is to have HOPE, an acronym she uses for Have Only Positive Expectations.

**A previous version of this story included information Kathy Perez shared in her conference presentation about the Learning Pyramid and rates of retention using various teaching methods which are in incorrect. We regret the error.*

SLA teachers see engineering as the perfect vehicle to get students practicing the transferable skills of breaking work down into manageable pieces, working together and learning from failed attempts. By introducing students to the built world and giving some simple ways to think about problems, they’ve also empowered students to design and build improvements for the physical school environment. And that freedom to make an impact has in turn attracted a more diverse set of students to the school’s elective advanced engineering classes.

The engineering programs at SLA’s two campuses are run by two teachers who used to work in the industry and remember exactly which skills they were lacking coming out of college and starting their first engineering jobs. “I felt like I didn’t know how to make enough stuff,” said Chris Pilla, the engineering teacher at SLA Beeber (a second campus that opened two years ago).

Pilla worked as a mechanical engineer at Lockheed Martin before switching to teaching. “I didn’t have enough experience working on and planning out a really big project,” he told educators gathered at the school’s annual EduCon conference. That’s what he tries to give his students in high school.

SLA Beeber is co-located with a middle school in a big old building that doesn’t have any of the open collaborative spaces teachers and students would like to have. But rather than seeing that as an insurmountable barrier, Pilla has incorporated the challenge of changing the physical spaces around the school into the engineering program.

They started by building a makerspace to house all their tools and provide workshop space for various ambitious projects going on around the building. “There was a huge advantage of doing that over paying an architect to design and build everything,” Pilla said. Every Wednesday afternoon from 1 to 5 p.m., Pilla and a handful of committed students worked on building the makerspace into exactly what they wanted. It took six to eight months and over 1,000 hours of manpower. But because students were so involved in its design and construction, they care a lot about keeping it neat and functioning, and want to help other students learn about it.

“It’s slow, but it’s tremendous for them because they know they’re building something that will be used by the school,” Pilla said.

The team intentionally built big glass doors into the makerspace so students walking by get curious about what’s going on inside and drop in to find out. The students who were most involved in constructing the makerspace are now so competent with the tools and protocols of the space that they are teaching assistants for Pilla. When students newer to making come in excited to take on a project, the old hands help them get up to speed on the skills. And a lot of those projects are about improving the school itself.

“We want to make sure that they can take control of the physical environment where they go to school,” Pilla said. That’s a radical idea, but it has been a tremendous way to engage students who might not otherwise be interested in engineering.

“It’s bringing in new people who might not have been into building the makerspace itself, but now they found a need in the building and are starting to get more involved,” Pilla said. Two girls who showed no interest in making or engineering before came to him with an idea to build a reading loft. They had identified a lack of quiet reading space as a school need and are now building it. They’re also taking engineering as an elective.

When kids are excited about what they can design and build, it makes it easier to excite them about more traditional engineering topics, too, Pilla said. Early on in his teaching, he tried to teach students about circuits. They gave up quickly and lost interest because it wasn’t connected to anything. But after they’d had a chance to prototype their own projects, build them, fail and try again, they had much more appetite for harder engineering challenges put forward by their teacher.

The SLA Beeber students and teachers have a lot of space to repurpose, which is both a lot of work and a luxury. At the Center City SLA campus space is tighter, but engineering teacher John Kamal still encourages his students to solve problems of design they see around the building.

“We’re just taking over any little places we can find,” Kamal said. Students noticed a hallway outside one classroom wasn’t being used for much, so they put up double doors and turned it into a storage room for some making equipment. Kamal and his students also converted a chemistry lab into a machine shop, putting the big equipment in the center of the room where the tables used to be and having students sit at the countertops in the back for times when direct instruction is necessary.

Using an engineering lens as a way of thinking about problem-solving and then letting students actually design and build solutions to those problems has made engineering a much more approachable subject to many students. Kamal said his goal has always been to draw more minority and female students into the discipline. Two years ago 70 percent of the engineering students were boys, partly because the courses were all electives. Now 41 percent of students in the program are women, up from 30 percent.

“I come from a family where everyone builds and what-not, but I was never really involved in it,” said Tiarra Bell, a senior at SLA Center City. Design drew her into engineering. She experimented with architecture and industrial design, but has really become passionate about furniture design. She now makes and sells her own furniture.

“It’s really cool because I’m a female and I’m teaching all the guys to do stuff,” Bell said.

**FOCUSING ON CORE SKILLS**

Kamal and Pilla meet with an advisory group of engineering industry professionals periodically to make sure their program is truly equipping students with the skills they’ll need to go into these fields later. When they ask industry experts the core skills required for good employees, no one mentions the ability to do differential equations. Instead, the qualities experts list look a lot more like what every teacher in every subject wants to see from students.

The experts say students need to be able to write, to find problems, to communicate, to Google, to understand constraints. They need to be creative, take thoughtful risks and have a “fearlessness to leap.” One project the SLA teachers have devised to help students work on all these skills is a massive Rube Goldberg machine with 70 moving parts designed by 30 people working together. There are lots of opportunities to fail on this project, but Pilla said he’s going to let the project continue until students have some success.

“I realized I wasn’t giving kids enough time to succeed after they failed,” Pilla said. He likes this project because it requires a lot of communication and careful design, as well as the ability to break a big project down into its many pieces and work on them step-by-step.

As students move into higher-level engineering electives at SLA (robotics, senior engineering, astronomy and space sciences, MakerSpace, electronics and programming), they get more and more control over the problems they’ll tackle, which is a challenge in and of itself. “We are so used to coming in and having our engineering teacher giving us a problem and a set of restraints,” said Javier, a senior at SLA Center City. In the advanced engineering class, the seniors run the whole class themselves, with Kamal playing more of a coaching role.

“We realized this is our class, it’s not his class, and he didn’t chime in until the very end to reflect,” Javier said. He’s found it to be good practice to sit down with peers and push one another to do the best work possible. Currently they’re working on designing a solar cooker that can be built out of materials in Madagascar, since it’s too expensive to ship parts.

“I don’t like engineering because of engineering,” Javier said. “I like engineering because of what it does for the rest of my life.” This multitalented young man is a self-described painter, writer and endurance runner. He says when he finishes a tough calculus problem that unlocks some part of an engineering challenge, it gives him confidence that he can finish a long run.

“To me it’s not about becoming an engineer in college or after. It’s about the critical thinking and the challenges and the creativity that comes with it,” Javier said. There was a collective sigh of longing and admiration from the educators in the room when he said that. What teacher doesn’t want his or her students to feel that way?

“We as educators are trying to develop whole people and that love of learning and that connectedness across the whole of life,” Kamal said. At both SLA campuses, engineering has been woven into the fabric of the school and has become a way for this community of people to come together and devise solutions that affect everyone.

And they’re taking it beyond the school walls. Pilla says his students’ next challenge is to transform a swath of concrete outside their school into a playground and community garden for neighbors to enjoy.

]]>Curiosity is baked into the human experience. Between the ages of 2 and 5, kids ask on average 40,000 questions, said Warren Berger, author of “A More Beautiful Question,” at the Innovative Learning Conference hosted at the Nueva School. Young kids encounter something new, learn a little bit about it, get curious and then continue to add on a little more information with each new discovery. Warren says that’s where curiosity happens, in the gap between learning something and being exposed to something new.

“Kids are lighting up their pleasure zones and getting dopamine hits every time they learn something that solves something they were curious about,” Berger said. He contends that questioning is a highly valued skill. Companies are looking for people who can ask deep questions that will solve real problems and lead to profitable solutions. Equally important, it’s up to an informed citizenry to ask questions about the world, policies and the actions of our government.

Luckily, kids are hard-wired for that kind of generative curiosity. Unfortunately, “right around age 5 or 6, questioning drops off a cliff,” Berger said. Paradoxically, when kids go to school they stop asking so many questions. “Children enter school as question marks and leave schools as periods,” Berger said, quoting Neil Postman.* But why?

There are a lot of understandable reasons why questioning drops off in school. Foremost among them is time. “Time really conspires against questioning,” Berger said. “In the classroom there often isn’t time to let kids ask their questions.” And really good, deep questions often take a lot of time to unravel — more time than a harried teacher trying to cover all the curriculum often feels she can afford. And while time pressure is a very real part of teaching, not making time for questioning says a lot about how valuable it is to us. People make time for the things they value.

But knowledge can also be the enemy of questioning. “As we know more, or feel we know more, we may be less inclined to question,” Berger said. Sometimes answers can close down other avenues of thinking or ways of seeing a problem, but that all depends on how teachers treat knowledge. When treated as a life-long endeavor, learning a little bit about something opens up space to learn more.

And of course there are social barriers to questioning. Many kids don’t see asking questions as “cool.” And the perception that question askers are suck-ups or dorks probably also comes from fear. Many people feel vulnerable admitting they don’t know something. They are afraid to offer a window into their inner world by wondering out loud.

These barriers to questioning are real and challenging, but there are lots of ways parents and teachers can work to make questioning a normal part of school and life. One of the primary ways adults can support questioning, Berger said, is to model curiosity and to value questions. Instead of asking a child, “What did you learn at school today,” a parent might ask, “What great question did you ask today?” Or, when a child asks one of those great, deep questions that gets at why humans are even here, parents could dive in and explore the question with their child.

“You don’t have to have the answers. You just have to have the interest,” Berger said. Instead of trying to close off questioning by providing a pat answer or a terse “I don’t know,” parents might say, “If you were going to start answering that question, where would you start?”

“We want their questions to be large and expanded instead of being diminished and eventually going away,” Berger said. That philosophy should apply to school as well.

**5 WAYS TO HELP STUDENTS BECOME BETTER QUESTIONERS**

**1. Make It Safe:** “I think this might be the most important one,” Berger said. Many kids won’t raise their hand in front of the whole class to ask a question because they’re shy or nervous. “Fear kills curiosity,” Berger said. “The two things do not exist very well together.” But a student that might be afraid to question in front of the whole group may be willing to ask questions in a smaller group or to write a question down. Teachers can help make small groups even safer by laying out protective rules like “no question can be edited or judged.”

“The key thing is it makes questioning the point of the activity, and that is rarely the case,” Berger said. “The point is always to get to the answer.” Asking good questions takes practice. The Right Question Institute offers protocols to get students questioning, but teachers shouldn’t expect kids to immediately be good at it.

**2. Make it Cool:** Berger suggests convincing kids that good questions lead to cool stuff and make the world a better place. Furthermore, people who ask good questions are cool people, even rebellious people sometimes. “The people who are really breaking new ground are the people asking questions,” Berger said. “Questioners are the explorers, the mavericks.”

And questions can make people uncomfortable, especially when they hit on something true. “If you are a questioner, you are going against the grain,” Berger said. “That could appeal to young people.”

**3. Make It Fun:** Turning questioning into a game can be a great way to make the process more lighthearted and fun. Frame the process as being a detective, solving riddles or puzzles. One possible game to get kids started is to take closed questions and turn them into open questions and visa versa. This helps kids really understand the difference and what makes a strong question.

Students could also approach the issue with “why” questions to dig into it, then start asking “what if” questions to open up their imaginations and finally “how might we” questions to begin coming up with solutions. “How might we” is a more invigorating and creative questioning tact that “how could we” or “how should we” prompts, which tend to have more judgment in them.

**4. Make It Rewarding:** Many students are used to empty praise from their teachers. When students venture a deep question, they commonly hear, “That’s a great question, let’s move on.” But an educator’s genuine interest in the question will be much more powerful than any praise.

Additionally, teachers can create structures in their classes to reward questioning. Perhaps there is a best question of the week, where students get to vote on one another’s questions. Or maybe there’s a bonus question on a test that is itself a question: “What question should have been on this test, but wasn’t?”

**5. Make It Stick:** Questioning has to be a regular part of the school day for it to become a student habit. The famous comedian George Carlin used to talk about “vuja de,” that none of this has ever happened before. He was joking, but he also credited his ability to look at familiar situations in fresh ways as a key to his success.

Educators could follow Carlin’s lead and spend some time one day a week looking at a common object or idea and pushing students to ask questions about it as if they’ve never seen it before. “If you can instill this habit of mind in kids, this is the key to success for innovators,” Berger said.

If educators can find the precious minutes to foster these habits, Berger believes it could go a long way to developing critical thinkers. “I know that often times it doesn’t feel like there’s room to do some of these things under the current schedules and demands, but I feel like what needs to be done is small acts of insurrection,” he told educators and parents gathered at the conference.

**Questioning Is About Power**

Feeling confident to question the systems of power around us is one of the key jobs of an informed citizenry. Kids need to learn during their time at school that they have the right to know, to challenge assumptions and to dig deeper. Fostering this mentality in students can be challenging for teachers who are often complicit in systems of control over students. But often when teachers open the space for these questions, value them and explore them with students, a deep trust is built.

“I also think questioning matters because questions open up a dialogue instead of shutting it down,” Berger said. He says it’s the honest, thoughtful, respectful questions that start really good discussions. And ultimately could lead to the equity that so many educators and students are striving toward.

It’s also important to note that questioning makes a student vulnerable, and every student has a different relationship and experience with standing up to authority. “It’s very possible that there could be some groups of kids who would be more worried about how questioning is going to make them look,” Berger said. “That kid has more at stake,” and teachers need to recognize that.

These equity questions are the next topic Berger wants to explore. One study he read showed that upper-income families encouraged questioning in school, while lower-income families told their children to fit in and not rock the boat.

“Just because they’re not asking a question doesn’t mean they won’t have them,” Berger said. He’s researching how people are making questioning safe for everyone. Ultimately, questioning and reflecting are the keys to self-growth, something educators want for all their students.

“It’s OK to ask ambitious questions about yourself, your life, and that you won’t have the answer right away,” Berger said. Often people don’t ask those kinds of questions because they’re afraid they won’t have the answer. But if questioning deeply has always been part of the learning process, perhaps the next generation of citizens won’t be so afraid to sit with those hard questions.

**An earlier version of this story did not properly attribute this quote to Neil Postman. We regret this error.*

This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.

“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the 2015 Innovative Learning Conference. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.

“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.

Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like Carol Dweck, Camille Farrington and David Yeager have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.

Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in a study of taxi drivers in London who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.

“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”

A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.

A recent study of students with math learning disabilities found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.

The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like visualizing math, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.

People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of how the brain works. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.

**GROWTH MINDSET AND MATH**

Increasingly, educators are buying into the compelling research showing that what students believe about themselves affects how their brains approach learning. Growth mindset is probably the best known aspect of this research, and many school leaders are trying to figure out how to implement growth mindset programs in their classrooms.

“More kids have a fixed mindset about math than anything else,” Boaler said. And it’s no coincidence that they feel this way. Teachers often believe their students can’t achieve at the highest levels, and in turn, students believe that about themselves. Plus, the tasks themselves communicate a fixed mindset.

“It is very difficult to have a growth mindset and to believe that you can grow or learn if you are constantly given short, closed questions with a right or wrong answer,” Boaler said. Instead, she recommends giving visual problems that provoke discussion and have multiple ways they could be solved.

She also says kids should not be grouped by ability or tracked into “advanced” or “remedial” groups. That common practice sends fixed mindset messages to students, both the “advanced” ones and the “low-performing” ones. Kids considered to be “gifted” suffer from ability grouping the most because they develop the ultimate fixed mindset. They become terrified that if they struggle they’ll no longer be considered smart.

Instead, mixed ability grouping can work if the tasks are open-ended and what Boaler calls “low-floor/high-ceiling” tasks that allow every student to participate, while allowing lots of space within the task for students to grow in their thinking.

Boaler has lots of example tasks on her website, YouCubed, and on the NRICH website.

**PUTTING IT INTO PRACTICE**

During the summer of 2015, Boaler invited 81 seventh- and eighth-graders from a low-income district near Stanford to come to a summer math camp focused on algebra concepts. She gave the students a pre-test and found that their abilities ranged from very low (getting 0 answers correct) to fairly high. Then, for 18 days she taught them math well.

The instructional program focused on mindset messages, was full of inquiry-based, low-floor/high-ceiling tasks, was visual and used mixed achievement groups. At the end of 18 days, when Boaler gave them another test they had improved on average by 50 percent.

“They improved because they changed their beliefs that they were not a math person to believing they were a math person,” Boaler said. After the course, students said they looked forward to math and saw math as a creative subject.

Administrators from the district came to observe partway through the camp and couldn’t tell who was a low achiever and who was a high achiever in the class. Boaler also makes it clear to the students in the workshop what she expects from them, and speed is not something she’s evaluating. Instead, they do norm building so that everyone knows how to appropriately work in groups, help one another and be supportive.

“If we don’t pay attention to those kinds of interactions, and kids are dominating, or thinking they’re smarter, then we’re really in trouble,” Boaler said.

Removing the time pressure from math is another important issue for Boaler. Neuroscience research out of Sian Beilock’s lab at the University of Chicago has shown that time pressure often blocks the brain’s working memory from functioning. This is particularly bad for kids with test anxiety.

“The irony of this is mathematicians are not fast with numbers,” Boaler said. “We value speed in math classrooms, but I’ve talked with lots of mathematicians who say they’re not fast at all.” But it is common for math teachers to call on the kids who get the answer quickly, reinforcing the idea for all students that rapidity is what matters.

**COMMON PUSHBACK**

Math education experts have been making the same case as Boaler for decades, and yet math education in the U.S. has not shifted much. Teachers often say they have to cover all the topics in the curriculum to prepare students for the tests they will be expected to pass, leaving them with no time for the kinds of open-ended, discussion-based math that Boaler advocates.

Boaler agrees with teachers that there is way too much to cover in the curriculum, especially because she finds much of it to be obsolete (don’t get her started on the textbooks themselves). “The most important thing we can give kids is to think quantitatively about the world and apply a mathematical lens to different situations,” she said.

In addition to teaching students, Boaler trains teachers in her methods. Often they go back to their classrooms and apply these theories, which means they aren’t covering every topic in the textbook, and yet their students do better on the standardized tests anyway. Boaler is not a fan of all the tests American students must take, but she says teaching math the right way deepens kids’ understanding of math in real ways that show up on tests, too.

Teachers and parents often push back against this kind of math. They wonder where memorization of math facts fits into the model, given the belief that kids must know their times tables to succeed in higher-level math. Boaler says that’s unnecessary. She is a math education teacher and has risen to high levels of math learning without ever learning her math facts. She has number fluency, knows how to manipulate numbers and understands concepts, but she doesn’t have her math facts memorized.

The Programme for International Student Assessment test (PISA), which is often used to compare achievement across countries, has a section about attitudes and beliefs. Those surveys show that kids who approach math as memorization are the lowest achievers in the world. “America has more memorizers than almost any country in the world,” Boaler said. The highest achievers are those who think about the big ideas and make connections.

Likewise, repetition of math tasks is not helpful to deep learning. The same kind of problem with different numbers does not improve understanding, Boaler said. What students really need is “productive practice,” approaching the problem from different directions, applying the ideas and explaining reasoning.

Boaler is on a mission to “revolutionize” how math is taught in the U.S. She has written several books to help teachers learn to teach with her methods, offers a free online course, and even gives away curriculum for teachers, students and parents on her YouCubed website. During one week at the start of the 2015 school year Boaler gave away five free math lessons, encouraging teachers to try this approach. She’s pleased that 100,000 schools tried the lessons, and teachers could see the difference in their students. A survey of students found that after the lessons and the growth mindset videos, 96 percent believed they should keep trying after making a mistake in math.

Boaler said a big problem is that math teachers themselves are math-traumatized. They came through a system very similar to the one in which they work. Elementary school teachers in particular often feel insecure about math.

“When they try math in these ways they get it, too,” Boaler said. “They can see this is much more valuable and enriching.”

]]>In turn, students regurgitate what they’ve been told, confident they’ve learned all the facts and unaware of the mysteries that remain unexplored. Without insight into the holes in our knowledge, students mistakenly believe that some subjects are closed. They lose humility and curiosity in the face of this conceit.

But our collective understanding of any given subject is never complete, according to Jamie Holmes, who has just written a book on the hidden benefits of uncertainty. In “Nonsense: The Power of Not Knowing,” Holmes explores how the discomforting notions of ambiguity and uncertainty affect the way we think and behave. Confronting what we don’t know sometimes triggers curiosity.

He wants students to grapple with uncertainty to spark their curiosity and better prepare them for the “real world,” where answers are seldom clear-cut or permanent. Whether exploring black holes or a Shakespearean sonnet, students should be comfortable challenging the received wisdom. There’s already a believer of the uncertain in science — Columbia neuroscience professor Stuart Firestein, who argues that “insightful ignorance” drives science.

“We’re much more certain about facts than we should be,” Holmes said. “A lot of this will be challenged, and it should not be embarrassing.”

If students can be made to feel comfortable with uncertainty — if they’re learning in an environment where ambiguity is welcome and they are encouraged to question facts — then they are more apt to be curious and innovative in their thinking.

Approaching knowledge this way is difficult for students and teachers, however, because ambiguity spurs unpleasant feelings. Indeed, studies show that the typical response to uncertainty is a rush for resolution, often prematurely, and heightened emotions.

“Our minds crave closure, but when we latch onto it prematurely we miss beautiful and important moments along the way,” Holmes said, including the opportunity to explore new ideas or consider novel interpretations. And teachers have additional challenges in presenting facts as fluid: appearing less than certain about their field of expertise can feel risky in a classroom of merciless teenagers.

But teachers who hope to inspire curiosity in their students, and to encourage tolerance for ambiguity, can take steps to introduce uncertainty into the classroom. Holmes offers several recommendations.

**Address the emotional impact of uncertainty. **“The emotions of learning are surprise, awe, interest and confusion,” Holmes said. But because confusion provokes discomfort, it should be discussed by teachers to help students handle the inevitable disquiet. “Students have to grow comfortable not just with the idea that failure is a part of innovation, but with the idea that confusion is, too,” Holmes writes. Teachers can help students cope with these feelings by acknowledging their emotional response and encouraging them to view ambiguity as a learning opportunity.

**Assign projects that provoke uncertainty.** One way to help students grow more comfortable with confusion is to assign projects that are likely to flummox them. Holmes identifies three techniques for doing so: inviting students to find mistakes; asking them to present arguments for alien viewpoints; and providing assignments that students will fail. “The best assignments should make students make mistakes, be confused and feel uncertain,” he said.

**Adopt a non-authoritarian teaching style to encourage exploration, challenge and revision. **Teachers who instruct with a sense of humanity, curiosity and an appreciation for mystery are more apt to engage students in learning, Holmes explained. “Those with an outlook of authority and certainty don’t invite students in,” he said. Also, when teachers present themselves as experts imparting wisdom, students get the mistaken idea that subjects are closed. “Teachers should help students find ways to think and learn,” he said. “The best teachers are in awe of their subjects.”

**Emphasize the current topics of debate in a field. **To give students a clearer sense of the mutability of facts, discuss the ongoing debates among academics and others on some “settled” subjects. Sharing what researchers, historians and theorists are arguing about now makes clear that questioning and challenging facts are what drive discovery.

**Invite guest speakers to share the mysteries they’re exploring.** In his class on ignorance, Columbia professor Firestein welcomes scientists across a spectrum of fields to talk about the unknowns they’re investigating. Chemists, statisticians, zoologists and others share with students the ambiguities that excite them, opening students’ minds to the vast unknowns waiting to be examined.

**Show how the process of discovery is often messy and non-linear. **Rather than present breakthroughs as the logical result of a long trek toward understanding, teachers can share with students how discoveries are often made: through trial and error, missteps, happy accidents and chance. Firestein describes scientific discovery as “groping and probing and poking, and some bumbling and bungling, and then a switch is discovered, often by accident, and the light is lit.” All the poking around in the unknown, he adds, is what makes science exhilarating.

**How Could This Look At Home?**

When Mollie Cueva-Dabkoski was growing up, her mother took her to the library every week to read stories together. When the storytelling ended, her mother asked questions that challenged the narrative and pressed Mollie to reconsider the protagonist’s motives, or to rethink the gender norms.

“She pushed me to question the world around me,” Cueva-Dabkoski said.

Cueva-Dabkoski, however, was troubled by all that she didn’t know. Raised by a single mother in San Francisco, and educated at an underfunded public school nearby, she worried that her ignorance about all manner of subjects would interfere with her ability to perform at college. Cueva-Dabkoski had always been curious and driven, but she doubted whether she possessed sufficient intellectual tools.

Awareness of the gaps in her knowledge spurs Cueva-Dabkoski to learn. So, she decided, “I taught myself how to be a critical thinker.” Today, she’s a junior at Johns Hopkins University, majoring in sociology and public health.

Though Cueva-Dabkoski laments what she calls the “product-driven” nature of higher education, she continues to challenge and explore, inside the classroom and out. As a teenager, Cueva-Dabkoski began to make a list of concepts she wanted to understand by age 20, and she continues to work her way down the list. Some subjects on that list? String theory, democracy in Burma, the history of Bhutan. How to explain her wide-ranging curiosity? “There are big gaps in my knowledge,” she said.

]]>It takes time to build up a strong inquiry-based teaching practice, to learn how to direct student questions with other questions, and to get comfortable in a guiding role. But when Laufenberg talks about what it takes, she makes it sound easy. We’ve broken her advice down into digestible tips for anyone ready to jump in and try for themselves.

**1. Don’t teach the content standards; help kids find their own path towards the information they need to know.**

Every teacher has a “bucket” of stuff she is responsible for teaching her students, known as standards. The best way to get students to understand and remember that content is to help them build their own path of questions towards the information they need to know.

“The brain is so primed for questions,” said Laufenberg, managing director of Inquiry Schools and a former 11th and 12th grade history teacher at SLA. “It learns better that way and remembers better that way.” Unfortunately, many educators and schools are so focused on achieving standardized outcomes that they don’t leverage the best tool at their disposal — students’ natural curiosity. School is full of questions, but for the most part those questions imply students should only know more about what teachers are asking them.

“At the end of it they may have consumed less content, but remember more of the sum total,” Laufenberg said. “And they end up in a better place.”

**2. Don’t tell students what they should know; create the structure for them to experience it on their own.**

“Inquiry at its best happens when the teacher is doing very little other than creating the architecture for the experience to happen,” Laufenberg said. “It’s asking the first question, putting up the provocative primary document or playing the two minute video.” After that, the room should be full of kid questions. And if a student gets truly stumped and asks for help from the teacher, her job is to ask another question that pushes the students’ thinking forward or raises new questions for the student to investigate.

Laufenberg has worked with well-intentioned, hard working teachers all over the country to infuse more inquiry into their teaching. Many of them find this model destabilizing because for a long time they believed their job was to teach content. To make inquiry-based learning work, teachers have to instead become experts at listening to how a student is thinking and then ask the one question that will “un-stick” the students’ thinking and set them off and running again.

“You know it’s happening when there’s very little telling of things, but rather leading of questions and experiences so the students discover those on their own,” Laufenberg said.

**3. Use class time to make connections between pieces of information.**

Especially with AP classes, students are motivated or else they wouldn’t be there. So give them a list of questions, tell them what to study and let them do so outside of class. They can use the textbook, the Internet and many other sources to find that information more efficiently and effectively than a lecture.

“Inside of class, use that time to make connections between information,” Laufenberg said. After all, what good are facts if they aren’t connected to anything else? “Give them [students] compelling things to do that have them analyze and talk to each other, and grapple with the difficulty of what’s going on in whatever it is you happen to be teaching. But stop using your minutes in class to just tell them things.” Teachers have the tremendously important role of helping students make sense of the facts they’ve learned and see connections to other issues.

**4. Many kids struggle with reading, so hook them with the non-written word.**

When Laufenberg taught at Science Leadership Academy she had a student in her class who was an advanced analytical processor, a great critical thinker and a wonderful problem solver, but she struggled to read and write because of learning differences. Laufenberg wanted her to be able to engage with the class content at the high level of which she was capable, and not be limited by her second grade reading level. She developed the habit of introducing lessons with something visual so the student wouldn’t be left out.

“I needed to do this because there was an acute situation I wanted to handle, but what it was doing was inviting all the kids to the table with a level playing field of comprehension, not putting the barrier in front of them to start with, which is the written word for comprehension,” Laufenberg said. She would show students something interesting or puzzling, even using 90 second videos to grab their attention. This strategy got students wondering and gave them a little background so that even if they were doing the reading Laufenberg assigned, they came to it with their own questions.

“Your reluctant readers are more likely to make an attempt,” Laufenberg said, because they are curious to find the answer to their questions. Laufenberg would often try to give students the baseline information they need to know in the quickest way possible. “We would background build, but it wouldn’t be, ‘I’m going to tell you a few things today,’” Laufenberg said.

If a complex idea can be imparted through a short video or other means, Laufenberg uses it so the majority of class time can be spent diving into deeper questions and analysis. Laufenberg always got at the background information through questions; she never just told students information.

“It isn’t less reading; it’s less reading of the least interesting information to yield the more in-depth reading and invested reading,” Laufenberg said. She still requires students to read, but if they aren’t reading for the background information then they can be engaging more complex and interesting texts.

**5. Stop giving struggling kids the most boring version of the work to repeat over and over again.**

“We do a really interesting thing in American education; when kids are struggling with something, we just give them the most boring version of it and more of it, over and over and over again,” Laufenberg said. There’s no way that tactic is going to get students excited about the subject they struggle to understand.

Math teachers commonly say they have to get through some basics in order to get to the interesting content. But if students aren’t interested in knowing, they’ll never get to the good stuff. “Getting kids to understand that math is not just computation, that math is this whole other thought process and way of thinking about the world, and really trying to understand the bigger picture of math,” is the key Laufenberg said. Kids have to care. “Give them a puzzle to figure out to then lead them towards the math that they need to know,” Laufenberg said. They need to figure it out on their own, or at least grapple with it to care.

She doesn’t think it’s that different from history. If the goal of teaching history is for kids to chronologically place events on a timeline, we’ve missed the full potential for the learning experience. If the purpose of math is only to compute, we’ve missed something.

And with reading, don’t give reluctant readers boring passages to read. Let them read whatever they want. No one wants to read things that are boring to them.

**6. Surprise students.**

Laufenberg would often start class by putting a primary source document up on the screen with no context. Students would come in and immediately get to work trying to figure out what the document was and where it came from. She says it was a great window into their thinking and questioning skills.

“Sometimes you can use really little projects to get their minds spinning on all the ways of knowing, and then model those for each other,” Laufenberg said. Not all the students will find the answer, but they’ll be curious to know how others did. Laufenberg calls activities like this “micro bursts of inquiry.”

**7. The traditional model of imparting knowledge isn’t working very well, so don’t be afraid to try out inquiry.**

“When people don’t want to do it I always tell them to pick the unit you know always falls flat,” Laufenberg said. “You’re not going to lose; they’re already not with you.” It’s a safe place to start because it can’t get worse and maybe some learnings will come out of the experiment that can inform other lessons.

**8. Find the “bend” in the outcomes and abandon the prescriptive path.**

Laufenberg recommends finding “the bend” by paring down the content to the most essential pieces and focusing on them thematically. That will help open up as many paths as possible for students to arrive at the big ideas that kids need to learn. When teachers assign a “project” that follows the pacing guide, has a definable outcome and which results in 30 assignments that all look the same, it’s not inquiry. SLA principal Chris Lehmann calls that “the recipe.” In a true inquiry-based assignment students will travel different paths to and produce different products, but learn along the way.

“In a non-inquiry classroom the kids will all walk the same path because the teacher has decided where everybody is going and nothing that anybody says all day long will alter that,” Laufenberg said.

**9. Indulge interesting student questions even if it doesn’t fit the pacing guide.**

Laufenberg has seen classrooms where a student asks a fascinating question that the teacher brushes off because there’s not enough time. Kids know when there’s nothing they can do to influence the direction of the lesson, a distinctly disempowering experience.

“Who that child is isn’t informing the path and that’s the most devastating part,” Laufenberg said. Listening to student questions and validating them by asking them of the whole group has the added value of building student confidence and highlighting the value of wondering.

**10. Approach the practice of teaching with inquiry and use that meta-practice to improve.**

“Most professional development has not asked the teachers to examine their own practice with inquiry,” Laufenberg said. But using inquiry to create inquiry-based practices is a great tactic to think through the essential questions teachers face.

]]>Math teacher Laura Kretschmar gave students a rubric with specific goals around collaboration, communication and instructions to use various functions in the program, but not a lot else. She’s intentionally giving them a lot of freedom to play with the program, create cool designs and figure out what the functions do.

“I think “y” means, like, going up,” says Juritzy Maldonado. “So to pull it up, I’m going to try to change the number.” She punches in 200 for “y” and watches the image she’s creating shift upward. Another group discovers that if they hit “repeat” multiple times, they can create a parachute-like design that they’ve figured out how to color in various ways. That wasn’t their original plan, but they’re running with it now.

“Pretty much everything we were doing is trying one-by-one and seeing what we got, and then we put them all together,” said Guadalupe Pena. She and her partner realize they haven’t used a crucial function to set “xy” but they’re not worried. “We still don’t know how to use [it] very well,” Guadalupe admits. “Since we’ve already got everything written down, we can take the risk to make it to see what it does to our parachute.”

This blind exploration using Turtle Art is part of a two-week deep dive Kretschmar is doing on the coordinate grid. She says it can be a tricky concept for a lot of kids, and it’s more fun for them to uncover the intricacies using Turtle Art. Having the context of their experience with the program makes the math concepts more relevant when the time comes to teach them. She also likes that while kids are exploring they’re working together, helping each other and building a visual reference point.

The Turtle Art project, and the concept of “doing” or “making” before any explicit instruction has been given, is part of the school’s attempt to shake up its teaching. Lighthouse Community Charter has to cover the same standard curriculum as district schools, so teachers have to choose carefully the times when they’ll spend a little more time and creativity on a difficult subject.

Student should stumble around a little bit noticing patterns and eventually walk away with some basics, says Aaron Vanderwerff. He’s the Creativity Lab and Science director at Lighthouse. He’s been coaching teachers on how to incorporate “making” into their curriculum when it’s appropriate. He says about 70 percent of the staff ask for help from the Creativity Lab each year.

“Core teachers are interested in trying to integrate this,” Vanderwerff said. “The concept of the coaching is that if we help someone with one or two projects, they may do more on their own.”

He runs workshops for teachers designed to give them the experience of learning through making and inquiry, so they understand how the framework can help their students. And it’s working. The high school physics teacher had students build a mousetrap car to learn about forces. Fourth-graders studying westward expansion built their own version of the Transcontinental Railroad, including engineering a way to get their trains over the mountains.

The school has built a makerspace that high school students use for robotics, a scientific inquiry class and even some art classes. Six years ago, Vanderwerff was the robotics class teacher. His success with a more hands-on, student-driven curriculum inspired the school to expand that work into the Creativity Lab and to incorporate “making” into all K-12 classes.

“We’re seeing that making really helps kids with that STEM (science, technology, engineering and math) piece of things if that’s something they’re excited about,” Vanderwerff said. While Lighthouse has only just recently graduated its first class of seniors, Vanderwerff and his colleagues were concerned as they watched other Oakland high school students attend college, encounter difficult STEM courses and give up.

In Lighthouse robotics and making classes, students work on the same project for six months. They naturally encounter obstacles, develop solutions and keep working. The class also gives students some hands-on experience with concepts they’d otherwise only learn about more traditionally. Suddenly, physics has a point, geometry comes alive and computer programming doesn’t seem so boring.

“Our goal is not to create more scientists and engineers,” Vanderwerff said. “It’s to leave doors open for kids.” He’s painfully aware that not many schools in the East Oakland neighborhood that Lighthouse Charter serves have makerspaces. The Creativity Lab and infusion of making into the curriculum schoolwide is a larger attempt to even the playing field and provide kids in this low-income urban neighborhood access to creative spaces.

“My students in their communities are not exposed to designers and engineers as much,” Vanderwerff said. His students have told him that his robotics class changed their plans for the future, not because he told them they should be an engineer or a computer programmer, but because they experienced the power of designing and making something.

“I would much rather push for this kind of curriculum in schools serving low-income communities than in other schools because I think it will help students to gain their own voice, and a lot of the kind of character-building aspects that are intrinsic in this, but also to be exposed to new possibilities for the future,” Vanderwerff said.

He’s disappointed that the maker movement isn’t more diverse, but says when he takes his mostly African-American and Latino kids to Maker Faire each year, they hardly notice. They are on fire with the ideas on display and proud of their accomplishments.

Vanderwerff is working with educators from around the country to promote making and design thinking in the classroom. He runs workshops open to public and private school teachers alike, hoping to spread some of these ideas beyond the likely suspects. The Creativity Lab has lots of project guides on its website, along with examples of student work.

]]>It was 2013. She was the new principal of the O’Keeffe School of Excellence, an elementary school on Chicago’s South Side that had been struggling for years. Finally, the school district had taken dramatic action by firing the principal, the staff and all the teachers.

That’s when Bankhead was hired. Her job was to turn a failing school into a successful one, with all the same kids, but an entirely new teaching staff that she got to choose.

Bankhead had a very clear idea about what kind of teaching she wanted to see at her school. She calls it “inquiry-based” teaching. It’s an approach, supported by research, that begins by posing questions to students rather than presenting them with facts or knowledge. It’s the opposite of the way she was taught.

“My teachers stood in the front and talked,” she says. “And that was it.”

To help the teachers at O’Keeffe learn how to do inquiry-based teaching, she gave them training. Lots of training. She set up workshops and sent them to professional development days.

But, it wasn’t working. She and her administrative team would visit classrooms, hoping to see all this great inquiry-based teaching. What they saw instead were a lot of teachers standing at the front of the room, talking. The teachers were learning about inquiry-based teaching at the workshops, but they didn’t know how to actually *do* it when they got back to their classrooms. So they fell back on what they remembered about how their teachers taught, says Bankhead.

This is a common complaint about the traditional approach to teacher professional development in the United States. Teachers go to workshops and professional development days where they might get great new ideas about teaching. But when they get back to their classrooms and try to put those ideas into practice, all kinds of questions come up. And the expert who led the workshop isn’t there to help. Often, there’s no one to turn to for help.

Teachers in the United States have been expected to go into their classrooms, shut their doors, and figure things out on their own.

Bankhead and her administrative team realized the typical American approach wasn’t going to work if they wanted to dramatically change teaching at their school. One of the O’Keeffe assistant principals had recently learned about an approach to professional development called “lesson study” in a class taught by a Japanese professor. They decided to get in touch with the professor, see if he could help them.

**Bringing Lesson Study to Chicago**

Akihiko Takahashi is a professor of math education at DePaul University. Before that, he was an elementary school teacher in Japan. He first came to the United States in the early 1990s looking for all the great approaches to teaching math that he and his colleagues in Japan had learned about from American researchers. When he couldn’t find these approaches being used in classrooms, he soon realized why: There was no lesson study in the United States.

Lesson study is a form of professional development Japanese teachers use to help them improve and to incorporate new ideas and methods into their teaching.

“If there’s no lesson study,” Takahashi says, “how can teachers learn how to improve instruction?”

Here’s how lesson study works.

A group of teachers comes together and identifies a teaching problem they want to solve. Maybe their students are struggling with adding fractions.

Next, the teachers do some research on *why* students struggle with adding fractions. They read the latest education literature and look at lessons other teachers have tried. Typically they have an “outside adviser.” This person is usually an expert or researcher who does not work at the school but who’s invited to advise the group and help them with things like identifying articles and studies to read.

After they’ve done the research, the teachers design a lesson plan together. The lesson plan is like their hypothesis: If we teach this lesson in this way, we think students will understand fractions better.

Then, one of the teachers teaches the lesson to students, and the other teachers in the group observe. Often other teachers in the school will come watch, and sometimes educators from other schools too. It’s called a public research lesson.

During the public research lesson, the observers don’t focus on the teacher; they focus on the students. How are the students reacting to the lesson? What are they understanding or misunderstanding? The purpose is to improve the lesson, not to critique the teacher.

In the United States, we tend to think that improving education is about improving teachers – recruiting better ones, firing bad ones.

But the Japanese think about improving teaching. It’s a very different idea, says James Hiebert, an education researcher at the University of Delaware who has written about lesson study.

“Everything we do in the U.S. is focused on the effectiveness of the individual,” Hiebert says. “Is this teacher effective? Not, are the methods they’re using effective, and could they use other methods?”

Hiebert says to improve education in the United States, we need to shift from thinking about how to improve *teachers* to thinking about how to improve *teaching*. Lesson study is one way to do that, he says.

**Lesson Study at O’Keeffe**

Akihiko Takahashi now helps run an organization called Lesson Study Alliance that helps American teachers, mostly in Chicago, learn lesson study. One of the schools is O’Keeffe.

I visited O’Keeffe in January 2015 to talk with teachers about their experience with lesson study and to see a public research lesson.

One of the first things to understand about lesson study is that it’s a long process, kind of the opposite of the one-day workshop American teachers are used to. Teachers come together to identify a problem they want to solve. Then they spend months doing research and planning a lesson.

I spent most of my time at O’Keeffe with a group of three teachers who had been working together as part of a lesson study group since the previous summer. Angela Flores and Melissa Warner teach third grade. Wanna Allen teaches fourth grade math and science.

When they first came together to identify the teaching problem they wanted to solve, they had several things on their mind. One, they knew the overall goal for the school was for teachers to work on inquiry-based teaching. Two, they were thinking about the Common Core. That’s a set of new education standards that lay out what kids should know and be able to do in each grade. Teachers at O’Keeffe – and across the country – are still figuring out how to teach the standards. Lesson study, they thought, would be a good way to do that.

“I’d rather struggle together than struggle by myself,” says Flores. She liked the idea of lesson study right away.

Flores, Warner and Allen decided to plan a math lesson that would focus on the third grade Common Core math standards for geometry. They noticed that kids often struggled with understanding how to find the area of a shape. Memorizing the formula “length times width” wasn’t a problem for many of them, but they didn’t seem to understand what the formula meant. If they were asked to find the area of an odd shape – a parallelogram or a few rectangles put together – kids often had no idea where to begin.

It took months of planning and consultation to come up with a lesson plan.

“It’s a lot of meeting after school,” says Warner.

That gets a laugh from her colleagues. They don’t get paid for this extra time. Their principal, Bankhead, does arrange for subs to come in occasionally to free them up for planning. But for the most part, doing lesson study requires teachers to be willing to work at night and on weekends.

“The pay is in the results,” says Allen. “You’re getting better as a teacher.”

Warner says lesson study has helped her think about teaching in a new way.

“It was about me before,” she says. “It was like, these are the things I’m going to teach you, and this is my end result.”

She was more focused on whether kids could demonstrate what they’d learned on an assignment or a test. She was less aware of how kids were actually learning.

Lesson study helps you “get into new habits as a thinker, and as an instructor,” Warner says. “And I see such a difference in my kids because of it. I feel like in the past, if my kids got an unfamiliar problem, they would just shut down, not know what to do. Now everyone’s creating a solution, and then we’re ready to talk about it.”

Lesson study is a welcome change from the old way of doing professional development, Warner says. It’s no longer “you going back to your classroom and stumbling around with an idea.”

Now, she says, there’s someone to give you feedback and say, try it this way.

“It’s turned my practice around.”

**Results**

Teachers at O’Keeffe haven’t been doing lesson study long enough to know what kind of impact it’s having on student learning. Other schools in Chicago that have been doing lesson study have seen test score growth, but there’s no way to know for sure whether that’s because of lesson study.

There is some evidence that lesson study improves teaching. A recent review of research on professional development in the United States looked at 643 studies on approaches to improving math teaching. Only two of the approaches were found to have positive effects on students’ math proficiency. One of them was lesson study.

Jasmine Bankhead, the principal at O’Keeffe, believes lesson study is working at her school.

“I’m seeing much better teaching, and there’s an attitude in the building that we’re all in this together,” she says. “That’s what we needed here. I know that as I plan and budget that I have to make room for this type of collaboration in my school, so that my teachers can continue to grow.”

Catherine Lewis, an American researcher who has been helping teachers in the United States learn lesson study for 15 years, says she recently asked one of the teachers she’d been working with, what’s the biggest change with lesson study?

She says the teacher told her, “The talk around the water cooler has really changed. We used to hide it when we had a failure. And everybody has failures in teaching. But we used to hide them. And now, we’re perfectly comfortable saying, ‘You know, I don’t have a good way of teaching division with remainders. What do you do? Can I come see it in your classroom?’”

It’s hard to know how many teachers in the United States are doing lesson study. There’s no official count. Lewis estimates thousands of teachers are doing it.

There’s even a whole state that’s trying it: Florida, which got a federal grant in 2010 to encourage its schools to adopt lesson study.

But lesson study can be challenging in American schools. There are practical challenges, like finding time for teachers to plan together and watch each other teach. Japanese teachers have this kind of time built into their work schedule.

And there are cultural challenges. The organizing principle behind Japanese lesson study is that the best ideas for improving education come from teachers. It’s a bottom up kind of approach.

In the United States, education improvement tends to be top-down.

“The American approach has been to write and distribute reform documents and ask teachers to implement those recommendations,” says Hiebert.

Lesson study flips the script. It’s one of the reasons so many American teachers who try lesson study like it. But it’s also why lesson study can be a fragile enterprise in the United States. There are plenty of stories about educators who start lesson study, then a new principal comes in with a different idea about how to do things, and lesson study falls apart.

Another challenge for lesson study in American schools is the fact that it’s a long and intensive process.

“We are so addicted to quick fixes,” says Hiebert. “If it doesn’t fix things in two years, it’s not worth it.”

We have this attitude about teachers too, he says. Research shows that teachers in the United States improve the most early in their careers, but after about three to five years in the classroom, they’re about as good as they’re going to get. If you’re not a great teacher after a few years, you might as well quit or be fired. That’s the thinking in the United States anyway.

But in Japan, you’re not considered an expert teacher until you’ve been in the classroom for at least 10 years. The Japanese take teacher learning seriously, Hiebert says. They believe teachers will improve if they work in a system that values improvement.

The United States needs that kind of system, he says.

“We have an education system that is always reforming, but not always improving.”

*Emily Hanford is an education correspondent for American RadioWorks, the national documentary unit of American Public Media. Check out the American RadioWorks website for a more in-depth version of this article. You can also read other articles about teacher learning and listen to the accompanying radio documentary program. American RadioWorks hosts a weekly education podcast available here. *

A question I’m asked often is, “Where should a teacher begin?” Should teachers just let students go or is there a process to good student-centered inquiry? I’ve reflected on this a fair amount, and I think small strategic steps are the key. I think letting students “go” without any structure will likely create failure, especially if students haven’t spent much time collaborating. Skills need to be modeled.

Many teachers have likely engaged in some type of inquiry or project-based learning, but with frustrating or dismal results. I hear things like, “students weren’t on task,” “one student bossed most of the kids around,” “the end product wasn’t very good,” and many more. I’ve had these same experiences. What I’ve come to realize when I see these “behaviors” for lack of a better term, it’s likely telling me students are missing skills, or a structure to help them through the learning process. It’s my job to ask kids questions to find out what’s really going on.

When I start with a new group of students, the design is tight. Choice is given, but I often pick the topic and options for student voice. I model skills like collaboration, thinking out loud about my learning, and explicating integrating tech and why it’s being used. I also add particular group activities that help kids develop these skills, and use rubrics, like those found on the Buck Institute for Education website, to help them assess their own ability to collaborate, etc.

I’ve also discovered I need to teach the difference between collaboration and cooperation. Most students have been taught to cooperate. “Play nice in the sandbox.” Collaboration is an entirely different thing. Many adults don’t know how to collaborate well.

**1. START WITH ONE UNIT**

Start with creating one inquiry unit in one subject. You can jump in and change everything at once like I did, but that’s slightly crazy. Instead, if you design one unit in one subject, at the end of each day, or week, you can analyze what worked and what didn’t. While teaching doesn’t always leave a lot of time for luxuries like reflection, it really is the key to figuring out inquiry learning, and as the teacher, it’s one of your most important roles.

Sometimes you may not understand why certain things aren’t working. Ask your students. I’m often surprised by how much they know and how adept they are at articulating what they need.

Two of the best resources I’ve found for creating an inquiry classroom are Carol Kuhlthau’s work and Alberta Learning’s Guide to Inquiry Learning.

If you don’t know how to create an inquiry classroom, ask me. I’m happy to help. You can begin by posting comments here. If you need resources, I can probably point you to some. Over the past year, I’ve had the opportunity to email, Skype and, if distance allows, have teachers, administrators and superintendents visit my classroom to see what we do.

**2. TALK ABOUT LEARNING**

Talk to your students about their learning — a lot. Especially in the beginning, I talk to my students about why my classroom is structured differently than every other class in our school. I show them Ken Robinson’s talk about how the 20th century school system doesn’t really prepare students anymore. I also show them Chris Lehmann’s TEDx talk emphasizing how education is broken and Karl Fisch’s Did You Know?

I tell my students that essentially I’m preparing them for jobs that don’t currently exist, that will use technology which hasn’t been invented yet, to fix problems we’re not currently aware of. They get the point. It’s about developing skills and habits of learning, and we use content to do that.

But I also talk to my students about stuff like how their brain works, and how neural connections need to be made. That often, in order for students to learn something new, it needs to be attached to things they already know. Just before the recent break, during the last week of school, we talked about cognitive dissonance and Vygotsky’s zone of proximal development. They like to know there’s a reason for the way they feel when they don’t “get it.” And they like to know that everyone’s zone of development is different. In fact, they were amazed to find out everyone’s brain is different.

And, yes, I use the big words. I simply explain what they mean. I don’t use them to sound smart. I use them because it makes my students feel smart; most of our society doesn’t treat our students like they’re capable of understanding or doing much. I do.

**3. MAKE TECH WORK FOR YOU**

Embed technology in ways that are authentic to the learning process. The first tools that I teach my students are Google Docs, Diigo or Delicious to bookmark their research, and Symbaloo to house their tools.

Experience has taught me that the first day I introduce a class to Google Docs, we will get nothing done. To them, it’s the most amazing thing ever. They usually spend most of the class typing back and forth to each other in the doc. No big deal. However, eventually, my students open Google Docs without me telling them to. I have students who literally use them for every lab, essay, and assignment. And the ability for a group to work on and edit the same document at the same time, more than makes up for the initial class we lose.

The social media tools we used to show our learning in our slavery unit seemed like the most natural and logical tools to use. As a learning community, we want our learning to extend beyond the four walls of our classroom. So we have a discussion, or likely multiple discussions, about what that should look like. We also want our projects to have “real world” implications. What’s more real world than advocacy against modern-day slavery using social media?

Essentially these are the two criteria we use to assess the product we’re going to create. How do we extend our learning beyond our classroom — and how can what we do here make a difference to the real world? Our tool selection is guided by the answers to these questions.

**4. EXPECT TO HIT THE WALL**

Remember that inquiry learning is an emotional process. Each stage of learning has specific emotions attached to it, and at some point, you and your students will likely hit the wall. That’s normal.

I’ve found that we need to talk more as an inquiry class. My role is to be well aware of how my students are doing emotionally, especially when we’re dealing with a weighty, overwhelming topic like slavery. While this may not matter much in a traditional classroom, it can completely blow apart a community learning through inquiry.

I won’t promise you that any of this will be easy. It’s not. You’ll likely have days when you wonder why you ever started it. But trust me, it’s worth it.

*This article originally appeared on Shelley Wright’s blog Wright’s Room where she explores her experiences in the classroom and ruminations on the future of learning. Wright teaches high school in Moose Jaw, Saskatchewan.*

This struggle may come from a fundamental misunderstanding about the discipline and how it should be taught.

That’s the stance David Wees has arrived at after more than 20 years of teaching at many different kinds of schools all over the world. It has taken a long time, but Wees has stopped labeling student work with the word “mistake” and has started paying attention to what he can learn about how students are thinking, based on the work (right or wrong) they produce.

“I want to know the ways that they are thinking rather than the ways they are making mistakes,” said Wees, who now works as a formative assessment specialist in mathematics for New Visions for Public Schools, an organization supporting public school teachers in New York City. “My interpretation that they’re making a mistake is a judgment and usually ends my thinking about what they are doing.”

In that situation, it’s extremely tempting to tell the student where he or she went “wrong” and move on. But what does the student learn in that scenario? Not much, beyond how to memorize computational formulas, said Wees.

“My goal is for them to become the truthmakers,” Wees said. “I’m trying to build a mathematical community where something is true when everyone agrees it’s true.” To do that, he asks students to talk through mathematical ideas, struggle with them and give one another feedback. “A major goal of math classrooms should be to develop people who look for evidence and try to prove that things are true or not true,” Wees said. “You can do that at any age”

Fundamentally, Wees wants to increase the amount of thinking “at the edge of their knowledge” that students do. “There’s lots of evidence that what we think about is what we know later,” he said. “I want to increase the amount of thinking going on in math class.”

Wees points out that while practice is important, students are repeating an action with which they are at least a little familiar.

He wants students to struggle in the zone of proximal development, where they don’t quite understand yet but aren’t frustrated. When working in New York public schools, Wees found if he gave students problems to solve that allowed for different points of entry, all students could struggle together. One student might be more advanced than another, but if each could access some element of the problem, they discussed it together and either relearned core concepts or were exposed to more advanced ones.

For example, Wees asked his students to solve the Seven Bridges of Konigsberg problem. It goes like this: A river flows through the middle of Konigsberg, forming an island in the middle and then separating into two branches. The citizens of Konigsberg have built seven bridges to get from place to place. The people wondered if they could walk around the city in such a way that they would cross each bridge once and only once.

“The kids understood the problem and virtually all attacked it,” Wees said. “Some kids worked on it for weeks.” Wees posted it in the hallway and at one point almost all the ninth-graders were working on the problem. Students got tired of carefully drawing the bridges, river and city over and over, so they naturally began to abstract the map into something that looked like a graph.

No student solved the problem — in fact, the mathematician Leonhard Euler proved it was impossible. Wees showed his students Euler’s proof, and pointed out how similar their graphing was to his. Wees said kids were a little mad when they discovered there was no answer, but they enjoyed the experience and along the way realized that learning is about the process.

“Over time I tended to embed projects of various kinds because at the time I was thinking I needed to get them interested,” Wees said. “They weren’t interested directly in the mathematics itself because they’d experienced so much failure, so I was trying to get them excited.”

Slowly throughout his career, Wees began to see that projects could be more than just excitement builders — they could be the vehicle for teaching content and the assessment. And the range of mathematical ideas was much broader than he thought if he used his imagination.

“The range of mathematical ideas the kids struggled with were pretty wide,” Wees said. After working in inner-city schools, Canadian schools and international schools for expat kids in London and Bangkok, Wees has come to the conclusion that all kids make the same kinds of mistakes.

“It was clear to me that the mistakes in some cases were a function of the mathematics and the way kids think about the math, rather than whether the kid is rich or poor,” he said.

**MATHEMATICIAN’S LAMENT**

Over the course of his career, through trial and error, Wees came to see what Paul Lockhart describes in his essay, “The Mathematician’s Lament”:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

**KIDS ASK THREE KINDS OF QUESTIONS**

When doing his master’s in education technology and the pedagogy around it, Wees learned to categorize the three kinds of questions students ask and changed his teaching practice entirely. Kids ask questions: 1) to find out if they did the problem right; 2) because the teacher is standing near them and they can, and; 3) occasionally they ask “I wonder what if” questions, which show they are thinking about the math. Wees took to not answering the first two kinds of questions and encouraging the third.

“I went from really trying to answer questions and support them in that way, to really trying to think of questions that would support them to learn it themselves,” Wees said. He found himself often asking the same question, whether a student had gotten the problem right or wrong. He’d ask them to explain their answer or how they could check to see if they were right or wrong.

“I became better at having a poker face so I wasn’t communicating whether they were right or wrong,” Wees laughed. When students asked questions because he was nearby, he deferred them to their peers, who often explained the math quite well.

**THE TIME FACTOR**

Many math teachers will say a community of learners like Wees describes is a fairytale classroom with no time constraints and no standards to cover. They say their jobs depend on covering all the topics on the test and helping students correct their errors, not taking days to uncover the thinking behind that error. Wees acknowledges the limitations that many math teachers struggle with, but points out the way most people teach math now doesn’t work, so it could be considered a waste of time anyway.

“Whatever time people are putting in to teach mathematics is kind of wasted in many cases,” Wees said. “Are [students] learning anything that they can transfer, that they can use in other contexts? If they’re not doing these things, then I don’t know what they’ve learned.”

He points out students often did very well on the New York Regents test when teachers focused on teaching specific kinds of problems, but whether kids learned the full range of mathematics possible that year is another thing entirely.

Beyond time limitations, a broader problem is that many math teachers know only one way to solve the problems they teach. Even professional development often focuses on breadth instead of depth, with the result that many teachers carry the same fundamental gaps in math understanding as their students.

“We have generations of math-phobia,” said Laura Thomas, director of the Antioch Center for School Renewal. “A lot of teachers who teach math are second- and third-generation math-phobic, so our system is really calculation-based as opposed to applying in context.”

Thomas said it takes a person with deep understanding of both math and project-based pedagogy and coaching to effectively lead students through what is often a very messy process requiring students to use problem-solving skills to figure out solutions, rather than being told what skills to apply.

Wees is frustrated at how linear math learning has become. “The standards are a list of things the kids are supposed to do, not a list of things you have to teach,” Wees said.

In other words, many standards can be embedded in a problem so that students are exposed to lots of ideas in different ways. When teachers focus on clusters of standards as opposed to individual ones, “that kid who doesn’t get one idea on Thursday is going to get 10 or 12 other ways of looking at the idea in the unit,” Wees said.

For example, a teacher might give students this math problem: “I’m traveling 50 mph. How far will I have driven in 10 minutes?” This problem does not confuse students. They know what they are being asked and in discussing it they could hit many standards — multiplication, number lines, writing down possible solutions to think it through and fractions, to name a few.

“The kids get exposed to all of the standards every day in different ways,” Wees said. And more importantly, they’re having to think through the standards every day, leading to a deeper level of learning.

“You really have to understand math is a range of ideas and not individual standards,” Wees said.

When teachers are comfortable teaching in this more complex style, they are able to offer the multiple points of entry that allow for differentiation to take place — but in community, not isolation. If students are segmented out to learn only with the students “at their level,” some students will be in danger of never moving past fractions.

]]>Teachers at the public magnet school Science Leadership Academy use a project-based inquiry model of teaching in an effort to connect all subjects to students’ lives. Examining social justice issues by the numbers has proven to be one strong way teachers can connect student passions to math.

In one project, groups of three or four students were responsible for a written mathematical analysis of their topic, two visual representations of the data, an engaging public service announcement video explaining the data and a list of recommendations for how the issue could be addressed.

“The biggest part of this was finding this information and saying, ‘Now what do I do with it?’” said Zack, a junior at Science Leadership Academy who did this project in his sophomore year.

Zack’s group examined incarceration rates in the United States, with each group member looking at an aspect of the issue, like educational attainment or geographical location of prisoners. As they each researched their own part, they kept a shared Google doc with information they were finding, sharing relevant research with one another when appropriate.

One of the hardest parts of the assignment was taking research and framing it in ways that would be useful for their claims, Zack said. Without that step they couldn’t be sure they were accurately comparing different numbers.

“It turns out that just five states in the South account for 20 percent of the country’s total prisoners,” Zack said, a disproportionately high number for the population of that part of the country. When his research led him to that point, Zack decided to hone in on those five states to make his case.

“Simplifying the information made the info more digestible and applicable,” he said.

Zack’s group also found that the majority of prisoners are high school dropouts, most are living under the poverty line, and 33 percent of the nation’s black males will be incarcerated during their lifetime. The group recommended the government look at issues of bias within the criminal justice system based on this data.

“This was taking concepts we’ve learned, making them more complex or advanced, and seeing real world application for the math,” Zack said. “It’s important for getting students into math because you hear every day kids asking, ‘When will we ever use this?’”

Demonstrating mathematical concepts, like central tendency or odds, and probability, suddenly felt very real to students.

“We wanted to give students a lot of room to have choice,” said math teacher Brad Latimer of a project in his algebra II class.

Students chose social justice-themed issues that interested them and then used research and data analysis to prove how the topic connected to social justice.

Students had to document specific mathematical concepts laid out by their teachers in the assignment. While this was a group project, the assignment clearly states individuals are responsible for analyzing an aspect of the data in terms of central tendency. The assignments reads, “This should include a focus on mean, median, mode, range, quartiles, and IQR (interquartile range), and should also include at least two original percentage-based statements about your data.”

**SOCIAL JUSTICE AND STATISTICS**

Statistics is arguably one of the most useful math disciplines, since citizens encounter numbers proving claims everyday in the news and as justification for various political policies. That also makes studying statistics a powerful vehicle for interdisciplinary learning.

SLA’s statistics and “Science and Society” teachers teamed up to examine the differences between organic and non-organic foods, as well as their cost and prevalence in different parts of the city. Students learned about the science behind different growing methods and how they affect nutritional qualities of food in science, while doing a statistical analysis of food availability in Philadelphia.

Students in groups of three to five visited grocery stores, sometimes of the same brand, in different zip codes throughout Philadelphia. No student in the class could go to the same store.

“It increased our analysis because we had more data from around the city,” explained Adam, an SLA senior.

They had to look for and note the prices of the organic and non-organic versions of different food items their teacher, Mark Miles, had selected. Students took selfies of themselves in the stores to prove that they’d actually gone. Each student was responsible for calculating and interpreting the 5-number summaries and IQRs, and means and standard deviations. They also had to draw and interpret box plots and histograms for all the group’s prices together, non-organic prices, organic prices and the difference between non-organic and organic prices.

“Growing up in Philadelphia, there were a lot more stores with non-organic because it’s cheaper,” Adam said.

He went on to note that after learning about the nutritional value of organics he felt it was unfair that poor people in his city didn’t even have access to products that might improve their health.

Even students who struggled in math were engaged in these social justice-oriented projects because teachers were careful to build in authentic choices that allowed students to investigate an area of interest.

“A lot of kids who struggle in math don’t see the relevance or they say they don’t care about that application,” math teacher Erin Giorgio said.

She found that even the kids who say they hate math grab onto these projects, and the best part is that their research leads them to ask lots of questions as they grapple with their data.

**OTHER SOCIAL JUSTICE IDEAS**

There are lots of ways to make math applicable to problems in the real world, but it takes creativity on the part of teachers and students. Giorgio will sometimes ask her students to analyze attendance data in Philadelphia based on the kind of school students attend: magnet, charter or neighborhood. As they notice things like the fact that attendance is much higher at magnet schools, they start asking more questions and talk about the reasons why that trend holds true.

“The end game is to get kids to recognize that math is important in their life,” Giorgio said.

Other ideas might include using physics to engineer a product that helps someone else or using geometry to investigate architecture in different neighborhoods or acreage of vacant lots.

The power of investigating social justice issues by the numbers lies in high school students’ passion for changing the world. Adolescents are becoming aware of their place within the wider world and many want to have a positive impact on it. Understanding how math will help them do that only makes them more prepared.

]]>Some teachers are skeptical about “student-driven learning,” suspecting that it’s really just another chance for unfocused social time. It can often be hard to see behind the jargon the careful planning and teacher support necessary to ensure that students not only stay focused, but also produce high-level work. Educators often wonder how students can all be working on different projects but acquiring the same skills. It may seem challenging to keep track of 30 kids investigating 30 different issues, but when inquiry-based teaching is done well, that chaotic swirl of ideas and needs is based on a strong foundation of planning.

Science Leadership Academy in Philadelphia is known for its dedication to inquiry, practiced through project-based learning with public school students. As a magnet school, SLA does have an application process, but many students are not coming from schools where they experienced inquiry learning before.

An example of this approach can be seen in Joshua Block’s senior social studies class. Recently, his students put democracy through the wringer, investigating American democracy not as a static system developed hundreds of years ago by the founding fathers but as a living, breathing expression of citizenship today.

To do this, Block first introduced students to the work of Alexis de Tocqueville, a French political thinker who visited the United States in the mid-1800s and wrote about his observations of American democracy. With de Tocqueville’s primary text as a guide, Block asked his students to research and argue three perspectives on American democracy that aren’t common knowledge and that resonate with them personally.

Students chose diverse projects, including social movements like Black Lives Matter, the effect of stereotyping, whether the American Dream exists, how the food industry affects people’s lives, the intersection of poverty and education, even juvenile incarceration. They analyzed their topics in a paper, made a multimedia presentation related to the topic, and designed and built a personal website to display their work.

“It’s not just the typical, watered-down way of looking at race in America, but a combination of their own experience, their reading and interviews with people in their communities,” Block said.

He believes the project is successful because students have so much choice within the broad parameters he sets. That doesn’t mean they can pick an easy argument and be done quickly. Block’s job as a teacher in this environment is to continually push the student to think from new perspectives, to find more sources and to use them more skillfully to bolster the argument. To do this, he asks lots of questions that spark them to push in new directions.

Stereotypes and American Democracy from Marlyn Mooney on Vimeo.

“The students are so much more connected to their work and passionate about it that they’re actually doing work that’s higher quality than they’ve ever done before,” Block said. He attributes part of this to the fact that students are presenting their ideas both through writing and a creative multimedia project of their choice. The projects live on a public facing website. The authentic, worldwide audience — coupled with students’ passion for their topics — means that they are hoping to sway public opinion with their work.

“I felt really strongly about all of the topics I’m talking about, mostly about poverty and education,” said Liza Cohen. She investigated various aspects of socioeconomic disparities in educational opportunities. The experience has her fired up to go to college, where she sees it as her civic responsibility to get herself educated and work on these issues.

**STANDARDS AND SUPPORT BEHIND THE SCENES**

While students had lots of choice in what to research and the kinds of textual evidence they would use to support their arguments, Block doesn’t give them an assignment like this and expect them to come back two months later with perfect projects. Instead, he has built in deadlines, peer review and class work time, so he knows where students are at throughout the process.

“I’m just assisting them to create this thing that we’ve all agreed is an important thing to do,” Block said. “It’s like we’re on a shared mission together to produce the best product possible.” When students turn in rough drafts, Block sits down with them and asks what they know they need to improve on. Then he’ll give them feedback he sees. At other points, students review one another’s work. Block says that often students give the same critique he would have offered. Students said seeing one another’s work helped them see where their own was lacking.

“We had to edit other people’s papers and give feedback, and you read it and realize it’s amazing,” said Marlyn Mooney. She loved learning about the topics her classmates were working on and continuing the conversation around big ideas beyond class. One part of her project investigated the food industry and some of its unhealthy practices. The issue has become a personal passion for her that she shares with her friends, perhaps more than they would like.

Block says in a class period he will spend about five minutes reminding students about upcoming due dates and things to keep in mind as they work. The other 60 minutes he spends moving through the classroom, asking questions to push students farther in their thinking, checking in on students’ progress and giving feedback.

Democracy and Education from Amani Bey on Vimeo

“I want to give feedback as much as possible while it’s in process and give students a chance to work on skills they are still developing,” Block said. He sees it as a mentoring relationship, and is certain that if he gave just one deadline the quality of work would be dramatically lower.

“It takes a lot of trust and confidence,” Block said, acknowledging that most teachers didn’t grow up with this kind of education. “If people miss some of the steps, then it’s not successful and they think it’s a waste,” he said. But it takes time to grow as a teacher in this system, which is why Block believes teacher mentorship is so important. He and his colleagues are continually tweaking projects to reflect lessons learned.

**STUDENTS TAKE PRIDE IN THEIR WORK**

“I’m most proud that I thoughtfully completed this entire project, and I walked away knowing 10 times more than I did before and feeling 10 times more involved than I did before,” Cohen said. “I feel like mentally, emotionally and physically, I got something out of the project.” She says that beats any grade.

“One thing that I’ve learned about myself is that I can articulate how I feel in a very solid sense and that other people will listen,” said Ron Harper. “I finally realized I have a voice and I can use it. And that feels good.” He focused on juvenile incarceration, as well as the power of social movements like Black Lives Matter. Harper says the research he did on his own made him sad, but that he was motivated to keep reading because he cared about the topics he chose.

Other students expressed their appreciation for a project that let them examine society and their place in it. Many students struggled to condense all their research into cohesive, clear arguments, but they often expressed pride both in their writing and in their multimedia representations of issues.

In a reflection submitted to Block once the project was finished, Molly Olshin wrote: “Research, media, and writing is not that difficult when you have the freedom to write on the subject you want. I think it’s important to give students freedom on writing and reading, because it’s hard to enjoy a subject you’re forced to research, especially if you don’t [like] the subject.”

Another student, Darya Nemati wrote: “I obviously learned way more about the three topics I was investigating, but I think I also was able to dig deeper and learn about myself as a learner. I was able to investigate topics that were outside of my comfort zone and with each chapter I was able to delve just a little deeper into my research and analysis. “

Students weren’t shy about talking about their challenges as well. Amani Bey wrote: “I struggled with slimming my idea down and at the same time writing a bunch about it. I think it made me stronger. To be able to write on an extremely particular claim and to expand on it and all its nuances have made me a great thinker.”

**BLOCK’S TAKEAWAYS ON INQUIRY**

- “This is not an assignment where there’s one right answer and where I have a specific vision of what the final product will look like,” Block said. “I give them the framework and students fill in the gaps. They do it through their own curiosity and creation.” That is a key takeaway for teachers interested in teaching with inquiry. If students are really allowed to bring themselves to their work, their final products won’t all come out in the same form. But that doesn’t mean they can’t all be evaluated using the same rubric and set of standards.
- Block thinks about what standards he wants the project to hit first and then builds backward from that point. In this project, students are meeting literacy and research standards through extensive individual research, lots of writing, crafting arguments, interpreting texts from both primary and secondary sources, and using that evidence to support their claims. Whether the student is looking at racism in America or the plight of undocumented people, those standards are embedded.
- Inquiry is based around the act of questioning. As the teacher, that means asking common questions like, “what is the main argument of your piece,” but it also means asking questions about students’ ideas and sources that push them to look behind what can often be a one-dimensional viewpoint.
- Structure long-term projects with deadlines (research document with quotes, rough draft, final draft, multimedia project), lots of formative feedback both from peers and from the teacher, self-evaluations and reflection pieces so that students can look back at what they took away from the work.
- Model good work throughout the process. That could mean holding up a student’s clear introductory paragraph or reading a de Tocqueville observation together as a class. “I’m breaking down the distinction between professional work and student work,” Block said. He wants students to feel that their voices are just as important as those they read in books. Reminding them that their work is part of a large conversation around these issues helps them see why their work deserves the time and effort they’re putting into it.
- Teaching this way is exhausting, but also incredibly rewarding. “It’s very grueling, but it’s also incredibly stimulating,” Block said, “I’m excited about their ideas and their work.”

Block says it’s amazing to see what young people create when they are given freedom to choose. He’s also always impressed at how the questions of democracy leak out of the classroom and into everyday conversations. Students come in and talk to him about what they’ve been reading or he’ll hear them discussing popular songs in new ways. Investigating democracy has taught them to look at their own world in new ways.

]]>Coverage of technology in higher education often stops at whether Massive Open Online Courses (MOOCs) can be an effective way to educate hundreds of thousands of students cheaply, or focuses on the newest app to help students track their classes and homework. Much of the technology marketed to universities targets administrative tasks, things like registering students or sites like Blackboard and Moodle that make it easy for students to check assignments and download readings. But especially in a seminar setting, some professors are using technology in ways that mirror some of the forward-thinking practices of K-12 teachers who are known for applying inquiry-based methods, accessing low-cost technology that’s easy to use and making the subject relevant to students’ lives.

**EXPANDING THE CLASSROOM**

Ken Koltun-Fromm is a religion professor at Haverford College, a small liberal arts college outside of Philadelphia (full disclosure: I attended Haverford, although I wasn’t a religion major and didn’t know Professor Koltun-Fromm as a student). He’s been experimenting with various ways to bring technology into his teaching when he thinks it could enhance the classroom experience.

When planning a course on modern Jewish thought, he wanted to capture the notion that this subject area is alive and still being developed, so he contacted colleagues in the field from universities around the country and asked them about their own research and writing. Each week he assigned his students to read a primary religious text recommend by the colleague, along with his or her analysis of that primary text. He then Skyped those colleagues into class to discuss their writings with his students.

“It was a way of opening up by bringing into the classroom modern Jewish thinkers who are engaged in modern Jewish thought,” Koltun-Fromm said. Students wrote their final papers about those same scholars’ work, which Koltun-Fromm sent to his colleagues. He then brought them to Haverford’s campus for a symposium and organized a breakfast where students could meet and discuss their final papers directly with the authors of the texts they’d investigated.

Koltun-Fromm paid for the symposium with a grant from the college’s Hurford Humanities Center, one of the many luxuries of a small private college, but everything else about the course design used very simple technology. The focus was on how technology could enhance human contact.

**PUBLIC TECHNOLOGY**

It may sound like Koltun-Fromm is a technology evangelist, but in fact, he has strict rules about the kinds of technology his students can use in class. He doesn’t allow them to bring laptops to take notes because he knows all too often they are checking email, browsing the internet or distancing themselves from the discussion in other ways. He’s much more interested in how technology can bring people together.

Koltun-Fromm received a small Teaching with Technology grant from the college to buy three iPads for a class he was teaching on visual and material culture. In one class session he asked students to find representations of Jewish imagery and screencast them for the whole class to see. “I wanted to think about what it would be like to have public technology, so you couldn’t hide, you had to share,” Koltun-Fromm said.

While discussing the images students found, Koltun-Fromm realized many students had no idea where their images had come from, which led to an interesting discussion about Google search and images within context. In fact, using Google helped students understand the reading they had been assigned for that class, David Morgan’s “The Sacred Gaze: Religious Visual Culture in Theory and Practice.”

“We realized that this methodological discussion we were having didn’t apply to Google images,” Koltun-Fromm said. Google flashes images at a user and there’s no time to engage in the kind of “seeing” the theorist described. The technology actually helped students understand the core concepts better, which gets to Koltun-Fromm’s basic framework for using these tools: “use technology as a mode of inquiry and as an object of inquiry,” he said.

**WHEN ONLINE DISCUSSION COMES BACK TO CLASS**

Koltun-Fromm has also experimented with asking students to post reactions to readings online that can then be brought back into class discussions. At first he tried to use the blog function on Moodle for this part of the class, but he quickly found that students associated Moodle with accessing and downloading assigned reading, not as a creative space. So Koltun-Fromm created another site with a layout that more resembled Twitter. His students were much more able to respond quickly and creatively on the new platform.

Asking students to respond to readings before class gave Koltun-Fromm an important window into what students were thinking. “Students who are shier sometimes use these blogs to articulate their voice,” Koltun-Fromm said. After reading their arguments he could draw quieter students into the conversation by validating their ideas and teasing them out more. The technology helped expand class discussion and include more students. This approach can also work in K-12 schools.

Another time, Koltun-Fromm wanted to help his students make connections between texts they were reading without relying on the linear format of a class syllabus, which usually proceeds week by week. Instead, Koltun-Fromm had students use Prezi, an online presentation tool that lends itself well to visualizing connections. Students demonstrated their visual literacy skills by making connections across texts, images and other assignments. It was a relatively simple way to get them thinking about how various texts speak to one another, no matter what form they’re in.

**TO FLIP OR NOT TO FLIP**

Professors of large survey courses are increasingly flipping their classrooms, putting the lectures online and using class time for questions or experiments. But Koltun-Fromm isn’t interested in that use of technology because he says it doesn’t change the fundamental assumption that the professor is the sole force of authority.

“Part of using the iPads in the classroom was to democratize the learning, to make the learning more active,” Koltun-Fromm said. By allowing students to find and project images they believed fit into the discussion, they each played a more constructivist role in facilitating class, allowing Koltun-Fromm to react to individuals and their ideas. He’s intentionally trying to subtly undermine his own authority, knowing full well that his students know who ultimately gives them grades. He finds the approach lets students take more control over the experience without affecting their respect for him.

In many ways Koltun-Fromm’s forays into using technology in his classroom are very simple: Skype, screencasting, Prezi, and online blogging don’t require extensive equipment or software, but they can enhance learning when used judiciously. The principle of using technology when it is appropriate and leaving it to the side when it doesn’t help fits the mentality of many teachers at all levels and disciplines. While small Haverford classes do not typify the large university experience, there are many ways professors could allow technology to influence their teachingto move more nimbly within institutions of higher education.

]]>