It is no coincidence that there is overlap with my own framework for nurturing a problem solving culture in the classroom. If our goal in education is to create the leaders and problem solvers of the future, we need to ensure that we go beyond content delivery and provide experiences where students learn these 21st century skills.

Recent events highlight the critical importance of making this a priority. Real world problems require complex thinking and sophisticated understanding of relationships. In my observations of how people attack problems, both in the world and in the classroom, I have seen four bad habits that get in the way of developing 21st century skills in any meaningful way.

Over and over again, when a problem arises, I have witnessed people responding almost instantaneously with their solution. Frequently the solution is presented as the only viable option, and no other possibilities are considered. Whether it’s students designing a device to protect an egg from a high fall (here’s one version I particularly like from NASA) or a community trying to prevent mass shootings, we jump immediately from recognizing the existence of a problem to implementing the intuitive solution.

I believe the way we deliver our K-12 curriculum promotes this kind of thinking. There are few opportunities for real problem solving. Instead, we present facts and algorithms as pre-packaged chunks for students to consume and later recall. We occasionally tell the story of how the facts and algorithms came to exist, but rarely let students experience that process themselves. When we teach history, for example, as an inevitable sequence of events instead of showing the messy and time-consuming complexities of making decisions and considering alternatives, students may believe that there must be one right answer to every situation. They will then either pick one from whatever happens to be instinctively obvious to them at the moment, or they will wait for someone else to show them the answer key.

We need to begin teaching students about the additional steps between and beyond recognizing a problem and implementing a solution:

- Clearly define the problem
- Analyze and understand the problem and its context
- Developing multiple solution alternatives
- Analyze the solutions and select one
- Implement or test the solution
- Evaluate the outcome
- Repeat the cycle

Related to our desire for instant solutions is our tendency to pick the options that involve the least personal risk or loss, even if they are the least effective. “I’m all for preventing global warming as long as I don’t have to pay more for electricity.” “I want an A on this paper, as long as I don’t have to give up my video games or quit the soccer team.” This artificially limits our options when we’re developing solution alternatives.

Teach students how to really analyze solution alternatives and recognize potential long-term benefits from a short-term risk. Show them how to identify and distinguish needs from wants. Help them learn how to do deeper research about the implications and consequences of the alternatives.

Because we have been trained to accept what we’re told without critique or analysis, we treat all media as a source of authority. In an age when anyone can publish anything, this is dangerous. One symptom is the proliferation of fake or satirical news articles shared as fact. No, a man playing the Pokemon GO app did not cause a major traffic accident. A more serious symptom is that many people appropriate opinions expressed on social and traditional media as their own. Psychologists have shown that when the people we connect with have the same opinions as we do, those opinions are significantly resistant to change. The nature of social media in particular promotes the sharing of opinions disguised as unambiguous fact.

Schools should spend more time teaching how to parse the difference between fact and opinion, and in analyzing opinions. Learning to question the source, recognize bias and hidden assumptions, fact-check information, and consider alternative explanations and possibilities will help students form their own opinions instead of merely adopting those of others

In order to recognize opinions and analyze them, it’s necessary to really listen to and understand other people. Too often we hear little of that they are really saying as we focus on formulating our own response. A great deal of conflict arises because we bring our own assumptions to a relationship and don’t truly understand from the other person’s perspective.

This week I observed this in a summer class I’m teaching. Two students who are collaborating on a project had two different ideas about what their end product should be. The first student described his idea about solving a particular engineering design problem. The other student interrupted him and explained his solution. When I asked the second student to tell me what he thought about the first solution, he just looked at me blankly. I probed further: “What specifically is it about his design that you think isn’t going to work well?” The blank stare continued for a moment, then he said, “Mine is better because….” The second student hadn’t listened well enough to make even a single thoughtful statement about the first solution.

If we spend more time understanding how the brain listens and teaching active listening strategies, students will begin to listen more deeply.

Addressing these four barriers to 21st century skill development will go a long way towards improving outcomes in schools. What other obstacles do you see in schools and in society that we should consider as we help students become better critical thinkers and problem solvers? Share your ideas in the comments.

*Image credit: Barrier by Rae Allen (https://www.flickr.com/photos/raeallen/27533649/).*

- “What do I need to do to get my kid into the gifted program in your school?”
- “You know how important it is in this community to have your child labeled gifted.”
- “I need my child in the gifted class because I don’t want her in classes with those ‘other kids’.”

Statements like these are part of the reason that gifted programs are controversial. So in places like Newington, Connecticut, schools are moving to eliminate gifted programs. While cost is a significant factor in these decisions, here are four reasons gifted programs are irrelevant in the 21st century….

]]>People give more than money: clothing, bottled water, food, and blankets come in to disaster areas from well-meaning donors trying to make a difference. Unfortunately, not all of it actually helps.

CBS Sunday Morning recently ran a story talking about what disaster workers call “the second disaster”: the relentless crush of stuff that flows in from these givers. Juanita Rilling is the director of the Center for International Disaster Information in Washington, D.C. She says it’s not just that there’s too much stuff. “Generally after a disaster, people with loving intentions donate things that cannot be used in a disaster response, and in fact may actually be harmful. And they have no idea that they’re doing it.”

The story gave an example of a hurricane relief effort in Honduras where donations included high-heeled shoes and winter coats. Rilling makes the point that often when people give it’s more about them than the recipients. They give personal items so they can feel like they’ve at least done *something*. But there is little thought about what’s actually needed by the recipient, and people rarely try to find out.

Professional development often feels like this. Every professional I know wants to grow and learn and get better at their craft. Unfortunately, PD is usually sent to us from a distant administration building. The people in that building, with the best of intentions, provide training that has no meaning to the teachers or benefit to students. But at least they’ve done something. Right?

What can a teacher do, though, besides gripe about the disconnect between what Administration seems to think is needed and what is actually happening on the ground? Here are two steps you can take to improve the situation (and some bonus advice you can share with your administrator).

1. Plan ahead

Just as it’s too late during a disaster to think about what might be needed, it’s too late during a terrible PD session to think about what you could or should be doing instead. Think ahead to what you need to learn in order to be more effective as a professional, and start developing a list of goals and needs. This should be both personal and collective. Create your own individual list, and also work with your colleagues to craft goals and needs for students and teachers in your building and your district.

2. Communicate with your administration

Well in advance, talk to the administrators responsible for planning and organizing the professional learning you want and need. Be sure to share your goals, but also demonstrate why your goals and needs align with the direction and mission of the organization. Remember that you aren’t a solo practitioner–if you work for a school or district, you have a responsibility to to the community of learners as well as yourself. Show your administration how meeting your needs also serves the whole district, and you’ll be more likely to get a yes.

3. Be ready to show results

Administrators need evidence of success. It isn’t that they are inherently distrustful or that they need to micromanage. Administrators are charged with managing precious, valuable, and often very limited resources. They need to know that those resources are going towards something that will make a real difference for individuals and the organization. So think ahead about what you’ll be able to show as a meaningful outcome. Over time, repeatedly bringing back results will build confidence and you will get more opportunities to own your own professional learning.

And a note for administrators…

The CBS story makes two points about what actually does work in a disaster to relieve needs: networks and money.

Juanita Rilling says, “For me, the network is key. Who has the knowledge? Where are spaces that goods can live if there’s a disaster? Who’s really well-connected on their blocks?” The parallel for professional learning is to allow and encourage your teachers to get and stay connected. If they have opportunities to work as an internal team and also to collaborate with others outside the building and district, the network will be better able to assess needs and manage resources than you can.

And if you’re going to give something in a disaster, Rilling says, give money. “Money sometimes doesn’t feel personal enough for people. The reality is, it’s one of the most compassionate things that people can do.” The reason money is best is that it’s liquid. It can go towards whatever the immediate need is at the moment. In PD, the most precious resource is time. Give it to your teachers in abundance, and let them decide what to do with it.

If there needs to be accountability, don’t micromanage how it gets used. Instead work with teachers to agree on mutually beneficial outcomes and support them in achieving them.

When we are proactive and plan ahead for meaningful and effective professional development, everyone wins. This is the perfect time of year to be thinking ahead to what will make summer and fall learning experiences better. What other strategies are working for you to avoid disastrous PD? What are you doing as a teacher or administrator to help everyone learn and grow and achieve meaningful goals? Share your thoughts in the comments.

]]>Fresh starts often involve fresh commitments. I propose that every educator should make the commitment today to take sixteen new steps in 2016 to deepen their knowledge, sharpen their craft, and stretch out of their comfort zone. In no particular order:

**Read every day.**I’m not talking about all those unread emails from central office, either. Blogs, picture books, journals, news, professional articles, YA literature, or poetry. It’s all input that helps shape and inform your thinking and provides cognitive fuel to keep your mind going.**Write every week.**Input isn’t enough. Process what you read and think about by turning it around and putting it into writing for a real audience. Everyone (that includes you) has something worth sharing, and most probably several somethings. Write a blog post. (If you don’t have your own, ask your favorite blog or blogger if you can guest post there.) Write an article for the district curriculum newsletter or for a professional journal. Start writing that book you have been planning in your head for the last eleven years.**Present to your colleagues.**Corollary to #2 is to share something in person. Did I mention that everyone has something worth sharing? Other teachers want to learn from you, too. Share a technique at a faculty meeting, run an inservice workshop, or present at a local or national conference.*This wine pairs very nicely with entree #7 below, too.***Learn a new tool every month.**Digital tech or low tech, find something new you don’t know how to use yet and start using it. Find other people who use it well and learn from them. Try stuff out. Break things. Then figure out how to fix them.**Listen to someone who disagrees with you.**And not just to plan your rebuttal. Really hear them. You will learn something new, and you’ll expand your awareness of different points of view in the world. It’s healthy for you and for your students.**Build something with your students.**Make something together that didn’t exist before and that none of you knew how to make before you started. Solve a problem, construct a toy, build a website, or create a work of art. Use your hands and your minds.**Attend an Edcamp.**They’re free, they’re everywhere, and they’re awesome. ‘Nuff said. (You say you’ve already been to one? OK, then become an organizer.) Oh, and since every Edcamp attendee is a potential presenter, you can check two items off your list on the same day.**Expand your network. And your students’.**The wider your circle, the more ideas you’re exposed to that can help you deepen your understanding. This works for your students, too, so help them connect with other students and other adults around the area and around the world.**Travel someplace new.**Whether you go across the ocean or across town, visit someplace you’ve never been. Repeat as often as possible, then use the experiences to feed #2, #3, #6, and #8.**Teach someone else’s class.**Pick a subject and grade you’ve never taught. Plan a lesson, a project, or a whole unit and teach it to a new group of students. Collaborate with the other teacher, or switch classrooms entirely.**Throw out grades and homework.**Examine carefully why you do what you do, and think about whether it is really as important as you think it is. If it gets in the way of actual learning, throw it out. Grades and homework are two things that most teachers do just because they’ve always done them. Take the time to really think about the purpose of your practice.**Teach something you’ve always wanted to teach.**Yes, you have time in your packed schedule to squeeze in one thing that you teach because you want to instead of because someone told you you have to.**Try Genius Hour.**To go a step further than #12, let your students tell you what they want to learn. Then get out of their way and let them learn it. Help where needed.**Get up and move.**Whether you add movement within your classroom routine or you get out and go somewhere else, movement is good for brains and learning. The change of perspective and environment helps keep things interesting and engaging.**Make a mess.**Learning should create dirty hands and piles of stuff and discarded scraps. Color outside the lines and put parts together differently than the directions say. Don’t clean up before it’s time. Or until you’re ready.**Remember why you teach.**Every day of 2016, you will be faced with things that discourage you and make you wonder why you should still bother. Be sure to take the time, every day if necessary, to remind yourself why you chose this. Your reasons may be different than anyone else’s, but they are still there if you look.

What would you add to this list? What personal steps are you going to take in 2016 to recharge, refresh, and grow yourself?

*Special thanks to Ann Leaness, Beth Still, KL Peters, Starr Sackstein, Steve Johnson, Seth Reichgott, Tony Baldasero, and Lisa Nielsen who contributed ideas to this list.*

Photo credit: “Take the First Step” by imanka

]]>I get it. You’re frustrated. You want to help your child with his or her school work. It’s supposed to make sense. After all, if an adult can’t grasp it, how is an 8-year-old supposed to?

So when it doesn’t make sense and you can’t help, something must clearly be wrong. And since the only obvious change was the Common Core, that must be the culprit….

]]>When I was ten years old, my family took a trip to Maine. It was the first time I’d been on an extended car trip out of my home state of Pennsylvania, and my father gave me a critically important job: *trip navigator*. It was my responsibility to help plan the itinerary and the route from suburban Philadelphia to Acadia National Park. My most important tool? A set of brand new maps of the mid-Atlantic and New England states.

I studied those maps for days before the trip, plotting out the best route to our destination. My planning was slightly complicated by the fact that we had specific stops we wanted to make: a day in historic Boston, and a visit to my mom’s college roommate. By the time we left, I knew those maps very well, and when the inevitable glitches arose along the journey, I was able to sort things out and get us back on track.

There were other times I didn’t have as much success. After I learned to drive, I frequently began to rely on my standard route to specific destinations. And there were more than a few times I had to go miles out of my way because I only knew those familiar roads to and from my house.

Teaching has developed much like my driving history. We’ve begun to rely heavily on purchased, packaged programs that give teachers the turn-by-turn directions without giving them enough time getting to know the map. This is highlighted in a recent editorial in Oregon’s Baker City Herald (*Solving 5J’s Math Problems*). The local school district had mandated use of EngageNY, a math program provided for free by the state of New York, and now teachers and parents were questioning its effectiveness. The editorial opens with an unapologetically blunt assessment:

“The Baker School District has a problem with its math curriculum. What’s not clear is how big this problem is.”

The editorial outlines three main problems:

- EngageNY may work well for some students but not others, but we have no idea how many or how well.
- The district implemented the program with little input from important stakeholders, resulting in a lack of flexibility for teachers who are expected to follow the math program verbatim.
- Flexibility needs limits, however. Teachers can’t be expected to juggle multiple programs every day to try and meet the needs of a diverse group of learners.

The newspaper calls for surveys of teachers and parents to find out more about whether EngageNY is working and to understand the problems more deeply. Without this information, the board and school district professionals are essentially driving blind, following the turn-by-turn directions in the program without having any way to know if they’re getting closer to their destination.

Before I recommend some possible paths forward to resolve these challenges, there are some additional problems we need to consider which weren’t directly addressed in the editorial.

**Flexibility is more complex than it first appears.** We tend to use the words “program” and “curriculum” interchangeably, but they are not synonyms. Let’s extend our travel metaphor further to better understand some of the components involved:

**Curriculum**is the map**Program**is the vehicle and turn-by-turn directions**Standards**define the destination**Instruction**is the journey**Learning**is the experience

A well-written curriculum gives teachers the whole map of the territory, while a program can give a recommended (or in the case of the Baker School District, mandatory) route through the territory.

The problem with rigid adherence to a program is when the unexpected takes place. On our trip to Maine, for example, we had a minor car accident in Boston that sidetracked us for a brief period of time. We might run into heavy traffic or a road closure. Weather can change the conditions or someone might need an unplanned rest stop. Wrong turns happen. If I’m following turn-by-turn directions and get off course, I may be completely out of luck. If I know the map, though, I can plot an alternative route that gets me back where I need to be.

Programs aren’t inherently bad, though. They provide consistency between schools and teachers, allow for a common language, and permit natural flow from one grade or school to the next because teachers know what to expect from the students they are receiving. Programs are also efficient in terms of time, teacher planning, and financial investment.

**Surveys won’t tell the whole story.** Surveys will reveal comfort more than effectiveness. Responses during early implementation will also be based on perceptions of self-competence rather than effectiveness. As Black and Gregersen pointed out in their book, *Leading Strategic Change: Breaking Through the Brain Barrier*, “Many prefer to be competent at the wrong thing than incompetent at the right thing.” Selecting a program and designing a curriculum can’t be a popularity contest. Surveys will also be filled out by those with the strongest feelings, skewing results. Feedback about EngageNY is likely to be conflated with overall opinions of Common Core, so those strong feelings will color perceptions.

**Meaningful change takes time.** Assessing the effectiveness of any new program should be ongoing, but you can’t make firm decisions about keeping or abandoning a path too soon. According to Michael Fullan (@michaelfullan1), a leading expert in systemic change in education, “Significant change in the form of implementing specific innovations can be expected to take a minimum of two or three years; bringing about institutional reforms can take five or ten years.”

Related to this, there is a very real and well-documented implementation dip that happens in any new endeavor. Those first two to three years will see a decline in performance, and to make long-range decisions based on those initial results means you will miss the opportunity to show eventual growth.

I propose three considerations as the Baker School District–and any other school facing a similar problem–looks for a way forward.

The Baker School District should invest heavily in embedded professional development and coaching in excellent instructional practices. And while a good program such as IMP will give you a head start on good instruction, emphasizing instruction will give teachers confidence to be more flexible when given the opportunity. This also means districts should emphasize fidelity to outcomes and goals rather than a program. Instead of checking whether teachers are using the right book or teaching the right page on the pacing guide, ask for evidence of learning that shows students are getting to the right destination.

It’s absolutely critical for a district to commit at the start of an initiative to persevere through it for the long haul. Educate stakeholders about the difference between what is comfortable, convenient or familiar and what is truly effective. Prepare everyone for the dip by setting reasonable short-term goals in addition to long-range ones, understanding that there is a predictable process of change. Use multiple ways of gauging effectiveness beyond annual test scores and surveys.

Give teachers both the vehicle (a well-designed program), and the map (a thorough curriculum) and empower them to make choices about the best way to get to the destination. Some teachers may choose to stick with the program. Others will find their own route. For those who want to take the road less traveled, provide a formal pathway for teachers to communicate their plans to leadership so that the district can maintain the common goal, common language, and ensure there is appropriate support for the teacher and students.

If you’re in a community that’s wrestling with these issues, here are a few questions for your own reflection and which may help you start this conversation with your leaders:

- What are your priorities and goals for student learning?
- How would you balance the need for consistency and fiscal responsibility with the desire for flexibility and professional discretion?
- When a teacher chooses to go “off route”, what are the important ways that your school and system can support that teacher to ensure students are learning? How should that be communicated to parents and the community?
- What other questions or concerns are raised from your unique perspective that we need to consider in implementing a new curriculum?

Educating hundreds or thousands of individual children in a school or district is a highly complex task, requiring a great deal of effort, energy, resources, and time. With adequate planning, thought, and teamwork, we can make it the trip of a lifetime.

photo credit: Should I ? via photopin (license)

]]>There’s a lot we can learn about teaching math skills from language arts teachers. For example, think about how we build writing fluency and grammar skills in language arts. Among other things, we certainly do these:

- Explicitly teach important skills and rules.
- Provide students with “mentor texts” and many examples of excellent prose to read and analyze; students who read a lot also write better.
- Ask students to review and correct examples which contain errors; this is often done in a “daily edit” format.

In math, most of our instruction traditionally focuses on the explicit teaching of math skills and rules, so we have that one down cold. Mathematical mentor texts are also used, though not as frequently, when we model solutions for students and share examples of worked problems….

*Read the rest of this post on Brilliant or Insane.*

photo credit: NCSSMphotos via photopin cc

]]>This post first appeared on the Education Insider blog at It’s About Time.

I’ve been a cook since I was big enough to kneel on a chair to reach the kitchen table. I remember my father teaching me to measure flour, a small cloud rising above the canister as I leveled the measuring cup with the back of a butter knife. A quick sift, cut in some ice-cold butter, roll and cut the dough into rings, bake for 15 minutes at 450°. I couldn’t wait to taste those fresh biscuits, the flaky, buttery crust dripping with even more melting butter. Before I’d ever really heard of Paradise, I already knew its official food.

Dad wasn’t my only cooking teacher. My mom was equally talented in the kitchen, and in a house with books in every room, cookbooks dominated the shelves. On weekends, Julia Child and the Frugal Gourmet were like extended family members.

In school, my first love was always math. Solving a difficult problem has always been satisfying. I started elementary school towards the end of the “New Math” movement, so we learned about set theory and base 8. By the time I entered high school, though, the “back to basics” shift was in full swing and my schooling was much more traditional.

Skip ahead to 1996. This was an important year for me, personally and professionally. A blizzard dumped 30 inches of snow on the Philadelphia region, closing schools for a full week. After four years of teaching third grade, I moved to fifth. Pennsylvania gave its very first state assessment. I turned 30. My second child, Timothy, was born, as was the Google search engine and the TI-83 graphing calculator.

Things were also beginning to change again in mathematics education. In this same year, the US Department of Education established the Mathematics and Science Educational Panel, tasked with evaluating the quality of K-12 mathematics programs.

1996 was also the year that Laurie Kreindler (@LaurieEDU) and Tom Laster founded IT’S ABOUT TIME™ (IAT), publisher of the Interactive Mathematics Program® (IMP). In the NSF-funded video above (Life by the Numbers with Danny Glover), an exuberant Danny Glover (@mrdannyglover), fresh from a successful run of Lethal Weapon movies, introduces IMP to an audience conditioned to teaching and learning traditional math. IMP, however, was anything but traditional. When this video was produced, featuring teachers and students from Philadelphia Central High School, IMP was only being taught to35,000 students in approximately 200 schools nationwide. IMP was a radical new way of thinking about high school mathematics instruction. With an emphasis on problem solving and understanding of deep concepts, IMP set a new standard.

“You might not realize it, but mathematics can unlock incredible power! Mathematics is a powerful tool for exploring life on earth and discovering our place in the universe. Why would anyone teach you what the tools are without helping you learn how to use them? That’s exactly how mathematics have been taught in this country for decades. All around the country, dynamic educators and innovative professionals are finding a way to make math make sense. Who knows, you just might discover that you are really pretty good at math because you’ve had the tools all along!” — Danny Glover (Life by the Numbers, 1996)

Today, my son Tim is a college freshman. It’s enlightening to compare how math instruction (and cooking!) have evolved in the years he’s been alive.

Twenty years ago, there were only a few chefs in popular culture: Julia Child, Wolfgang Puck, Emeril Lagasse. Their television shows were primarily about making sophisticated techniques accessible to untrained cooks using the equipment available in most home kitchens. Still, to admit you were a fan of cooking shows was to embrace the label of “nerd.”

Math in school was in a similar place. The “math nerds” didn’t hang out with everyone else. In fact, it was cool to proclaim that math just wasn’t your thing. And nearly everywhere, school math focused on computational fluency and getting the right answers.

But 1996 was at the cusp of changes in both fields. The NCTM standards had started a movement away from rote learning towards an emphasis on problem solving and application, exemplified byIMP. And the fledgling TV Food Network was beginning to change the way America thought about cooking.

Today, popular media and culture are filled with cooking competitions and festivals. Artisanal pickles and small-batch craft beers are mainstream. Chefs are celebrities, and they even have live touring shows. It’s cool to be nerdy about food.

Math sadly hasn’t made as much progress. Danny Glover told us about popular nineties icons bad-mouthing mathematics, and today Hollywood still hates math. Even when Hollywood shows it a little love, math is almost always the domain of the quirky genius or the mentally ill one. It’s math as spectator sport, still not something mere mortals should attempt.

The PSSA test in Pennsylvania brought with it a new kind of test item my colleagues and I hadn’t seen before on standardized tests: the open-ended question.

For me and my students, the most challenging part of these questions was the requirement to explain reasoning. For the first time, it wasn’t enough to pick the right answer out of a list. My students had never been asked to answer the questions “Why?” or “How do you know?”

So we developed some rudimentary techniques to start building those muscles. We gave students a challenging problem and asked them to work on it for a week. We encouraged them to work together and talk to each other. We spent time teaching writing and vocabulary. We even gave them problems to which we didn’t know the answer.

Although student engagement has today achieved buzzword status, it was hardly talked about twenty years ago. Nonetheless, when our students were working on complex problems, they were more deeply engaged. Our math class was being transformed into one focused on big mathematical concepts; without realizing it, we were inventing in our elementary classrooms what IMP was doing for high school math.

A similar transformation was happening in the world of cooking. As we turned to the 21st century, Alton Brown created Good Eats. Equal parts Julia Child, Mr. Wizard, and Monty Python, Brown did far more than just walk his audience through a recipe. He taught us why it worked. And did it in a way that was deeply engaging. My son Tim has probably seen all 252 episodes of Good Eats at least twice. Though he and I did spend a fair amount of time together in the kitchen, I suspect Tim learned far more from Mr. Brown.

For my entire school career, first as a student and then as a teacher, math class was fragmented into two-week segments. We’d spend two weeks learning, say, how to add three-digit numbers. One day we’d add without regrouping. The next we’d add with regrouping. Eventually, we’d take a test with all the kinds of three-digit addition mixed together.

The day after the test, we’d start a new unit: perimeter and area. Two weeks later: fractions. And so on. Many schools institutionalized this with scripted programs and tight pacing calendars. Thanks in large part to a ten-year experiment called Project Follow Through, direct instruction was touted as vastly superior to any other method.

This design was the epitome of what David Perkins calls “elementitis”: teaching a subject by breaking it down into its smallest elements and teaching them in isolation. The instruction is more manageable, but it sacrifices understanding for simplicity.

By far the biggest change in school mathematics is now happening as a result of the Common Core State Standards. Instead of fifteen topics to cover at breakneck speed, we now have three or four major topics each year. The explicit intention of the Common Core standards is to allow teachers and students to spend more time and go into more depth.

This does not go far enough, however. The groundwork has been laid, and if the next twenty years are going to be better than the last, it is time to reinvent mathematics.

While some schools may still choose scripted programs despite more recent evidence that direct instruction is ineffective, more reflective schools, like Etowah High School (“What Does Math Look Like in Today’s Classroom?”), will embrace this opportunity to rethink their instructional practices. Three things need to happen for school math to be relevant and meaningful in the twenty-first century:

One thing that hasn’t changed at all in the past twenty years: high school math classes still use the TI-83 calculator, even though today there are more options, like Photomath, Desmos, and Wolfram Alpha. Look at that Danny Glover video again: what was once state-of-the-art in video production today looks dated and even a bit cheesy. It’s well past time for us to have high def math.

Our reliance on old tools and methods just allows students to lose even more ground. “It’s not so much that maths education is worse than it was;” says mathematician Conrad Wolfram, “instead, real life is much more demanding and we’re running in the wrong direction to catch up.”

Wolfram’s approach to math is radical: stop asking students to do computations that can be done more easily by a computer. Although many schools may not be ready to dive head first into this deep end of the curriculum pool, we all must shift instructional time significantly towards mastering the few things that humans still do better than computers.

Any professional educator should be an expert in learning, and to do that you must understand the brain. We know far more today about how the brain learns than we did twenty years ago (as Sherry Fraser explains in an interview, “Sherry Fraser Doles Out Mental Ice Cream with Problem-Based Math.” Math educators must understand the mental processes involved in subitizing, number sense, and how the brain processes different kinds of learning experiences. They need to be aware of developmental and emotional factors and be able to recognize different causes for a student misconception. Teachers also must understand how to gain and maintain a student’s attention, when to shift gears, and how to design instruction so it aligns with the brain’s natural timing and patterns.

Connected with understanding the brain, math teachers must realize the importance of the entire learning environment on a student’s learning. Curriculum isn’t enough, nor is it adequate to rely on an instructional bag of tricks. Physical space has a significant effect on learning, as does the culture of the classroom. Math teachers must be intentional about not just designing lessons but designing the entire environment in which those lessons will take place.

The more I learn about cooking, the more complicated it becomes. Every new idea, technique, and ingredient carries with it a raft of other things to learn about making good food. I find the same is true for math: transformation will take time. We’ve come a long way since IAT and Danny Glover made that film. Programs like IMP are helping teachers do the hard work of thinking differently about what math class should look like.

But we still have work to do. Edward Begle said in 1972, “Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.” If he were alive today, I suspect he might say this is an understatement.

]]>It’s the time of year we celebrate scary things: ghosts, goblins, clowns, pumpkin spice (whatever that is). There is one creepy thing with the power to raise the hairs on necks everywhere and which rarely gets celebrated: math. This is a shame. Because the fear of math is not inherent in the subject. Rather, it’s a result of growing up in a school environment based on four terrifying myths about math:

- Math is applied computation, and being good at math is about memorizing increasingly complex incantations done with arcane symbols.
- Math achievement comes from implementing quality programs “with fidelity.”
- Problem solving is the last in a long series of teachable math skills.
- Some people just aren’t math people.

So let me turn on the lights and pull off the rubber mask to reveal the truth behind the myths.

The point of learning math is to understand how to make sense of problems and communicate with others about them. The structure, vocabulary, and grammar of a language shape the way people think. The rules of math are not just recipes we must blindly follow in doing computations. They are a precise and sophisticated way of structuring ideas and understanding solutions. Learning math is therefore far more than just memorizing the vocabulary and grammar. It’s about becoming fluent in the idioms and subtle features of math, which leads us directly to the next truth:

If we treated world language instruction the way we teach math, we’d have students drill on grammar rules and vocabulary for several years and the only sentences they would hear, read, speak, or write would be in isolated, contrived situations. There would be no conversations, no discussions, and no need to come up with your own thoughts in the other language. Now imagine one of those students was dropped into a foreign city for a week: that’s the annual state math test.

A better way to learn is to spend most of your time immersed in an environment where the language is used naturally. You still have instruction in the grammar and vocabulary, but it’s always done in the context of the way the language is really used by native speakers. In math, this means all skills, content, and structured algorithms are taught in the context of problem solving. Students solve problems every day, and they have abundant opportunities to collaborate just as adult problem-solvers do.

We often treat problem solving as just another isolated skill, as if it’s a simple checklist we can give students. “Understand, Plan, Solve, and then Check,” we tell kids. Just follow the recipe and you’ll solve every problem.

Just like communicating in a foreign language, problem solving is actually a complicated and sophisticated collection of skills and habits of mind. The power is not in accumulating all of the isolated problem solving techniques, but in being able to use them together in interesting and often messy ways. Like learning a language, learning how to solve problems is something that students do over a long period of time, and there is always more depth to discover from new and different kinds of problems they can encounter.

All of this presupposes that students are regularly encountering the right kinds of experiences to be immersed in problem solving. This doesn’t happen automatically or by accident. It has to be systematically designed. Ideally, a district team will create structures and provide resources to enable this, but an individual classroom teacher can also do this. The 5 Principles framework is a guide for developing aspects of your classroom and school culture that supports problem solving and innovative thinking:

**Conjecture**. Promote student inquiry and critical thinking. Never end with an answer, and always follow up with questions like, “How do you know?” and “Are there other ways you could solve that?”

**Communication**. Students learning a language need to use it. A lot. So give them opportunities not just to do rote computations but to express their reasoning verbally and in writing. Let them translate from the math language of symbols into English and back again.

**Collaboration**. Few problems are ever solved by one person working alone. More often, people work together to generate ideas and build on each others’ solutions. These are skills in themselves, and students can learn a lot about how to solve problems themselves when they see how others attack a problem.

**Chaos**. Let’s face it: there are few problems in life that are neat, organized, and prepackaged with exactly the right information needed to solve it. Students need to experience messy, complicated problems as often as they do the more straightforward ones we usually present them in math class.

**Celebration**. Yes, this means finding ways to make math enjoyable. But it is also about recognizing the positive value of mistakes and letting students celebrate the growth that happens afterwards. Students should have plenty of opportunities to recover from failure and show new learning.

Every educator has an obligation to create the right conditions for learning how to solve problems. Even if you don’t teach math, you can create a classroom culture that promotes the 5 Principles. Encourage your students to ask questions and communicate clearly, to work together and tolerate the messiness of real problems, and to celebrate not only the right answers but the learning that comes from mistakes.

What’s one thing you can do today to change the culture of your classroom? Share your ideas in the comments, and join the discussion at http://www.geraldaungst.com/5cmath. To learn more about creating a classroom culture that promotes problem solving and innovative thinking, check out my new book *5 Principles of the Modern Mathematics Classroom*.

My wife loves her vegetable garden. Michele will spend hours with her fingers in the rich soil, nurturing her tomatoes, zucchini, lettuce, green beans, jalapenos, watermelon, and sweet potatoes.

I love Michele’s garden too: I get to eat the produce it generates. The problem is, so do all of the local wildlife which didn’t get the memo from the township that we own this land now and they need to move along.

So over the past several years, I’ve built three fences for my wife’s garden. It’s not that they were bad fences, or that they didn’t stay up. It’s that the problem we were solving kept changing.

My first fence was awesome: nice, straight metal posts hammered into the ground at equal intervals around the garden’s perimeter; welded wire fencing tightly stretched between the posts and anchored into the soil to prevent forward-thinking groundhogs from digging underneath. And it worked exactly as intended. Those groundhogs stayed out all summer, as did the foxes, cats, squirrels, and other small mammals.

The deer? Not so much. To them, the fence merely defined the boundaries of a conveniently-located salad bar. We didn’t get much of a harvest that first year.

Year two brought fence number two. Taller posts. A second tier of fencing above the first one. This fence kept out the deer, and a lot more of the food made it into the house.

I thought I was done with fences, until this year when Michele said, “I want to expand my garden.” So with the help of my son, we added more fence around the new section, and the result is the verdant miracle you see in the picture above. And in case you were wondering: ratatouille made with fresh-picked zucchini is absolutely spectacular.

*What’s all of this have to do with math?* you ask. Well, the fence was built to solve a particular problem, and it wasn’t a solution I could look up in the answer key to a book. It wasn’t a nice, neat problem created by a textbook-writer either. It was messy (both literally and figuratively) and complicated, and the parameters kept changing as the situation evolved.

There was certainly computation involved in solving the problem: how much square footage we needed to have in the garden, how much fencing to buy, how far apart we’d put the posts. But it was more than that. Every version of the fence needed to innovate–granted, on a very small scale–in order to work in our unique situation.

I learned how to innovate by solving problems throughout my life. It’s not something I learned in school, at least not directly. But our modern world needs more innovators, and if we’re going to build these skills in our students, the mathematics classroom is a fantastic place to start. There are two habits of mind you can build into your math instruction to begin creating more innovators. I call them Conjecture and Chaos.

The human brain is naturally curious. We are wired to wonder. John Medina, molecular biologist, explains in his book Brain Rules that it comes from our need to explore our environment.

Babies are born with a deep desire to understand the world around them, and an incessant curiosity that compels them to aggressively explore it. This need for explanation is so powerfully stitched into their experience that some scientists describe it as a drive, just as hunger and thirst and sex are drives. (Medina, 2014, p. 247, emphasis mine)

Most math classrooms never capitalize on this curiosity. We hand students predefined and preprocessed problems for which there is one, single correct answer and then walk them step-by-step through the procedure for finding that answer. Worse, we wait until after they’ve drilled the skills to even introduce them to that problem.

Instead, try sparking their curiosity by introducing your students to an intriguing conundrum and then using it to introduce the skills they’ll need to solve it. You may be surprised at how many of your students figure the skills out on their own in the course of solving the problem.

When students struggle to get an answer, encourage them to guess and explain their partial reasoning. Never end with the answer to a question, either. Always ask your students follow up questions like these:

- Why do you think so?
- How do you know?
- Why did you solve it that way?
- What was hard about solving that problem?
- How did you overcome the difficulty?

Professional mathematicians exhibit this kind of curiosity all the time. It’s what propels them to explore new ways of thinking about the world and math. James H. Simons, founder of the National Museum of Math, turned his incessant curiosity into a billion dollar empire. And he did it by studying math.

Math class needs to be a place with a healthy dose of chaos. I’m not talking about the kind of chaos my students experienced during my first year as a teacher, when I struggled with even the most basic classroom management tasks. I’m talking about the messiness of real problem solving. Thomas Edison famously said, about the path to finding the right filament for his light bulb, “I have not failed. I’ve just found 10,000 ways that won’t work.”

One of the biggest problems with math instruction is that we show students the one way that someone else figured out would work, and then ask them to rehearse it to perfection. We never let kids experience the ten thousand ways that won’t work. I’m not talking about aimless, endless anarchy here, but we can’t expect kids to learn to cook without making a mess in the kitchen.

Terry Tao is another world-class mathematician, one who embraces the Chaos of real math work. When he tackles a complex problem, he never takes a straight line from problem to solution. The work usually involves weeks or even years of false starts, failed attempts, and flawed logic.

If one of the world’s greatest mathematicians is a messy problem solver, why wouldn’t we let our students be messy? When students struggle to solve a challenging problem, don’t be too quick to come to their rescue and provide them with the next step. Focus on the process.

One way to do this is to give students the correct answer to the problem up front. Now there’s no pressure to find an answer. Instead, the challenge is to figure out exactly how someone got to that answer. Another strategy is to use lots of non-routine problems. Consider this one:

If Citizen’s Bank Park were completely filled with popped popcorn, how long would it take the Phillies to eat it all?

Given the Phillies’ level of success during the 2015 season, perhaps we ought to seriously consider filling the ballpark with popcorn. But I digress.

The point here is that there is no way to be certain about the answer. Students need to defend and explain their thinking, moving them towards deeper thinking practices that will serve them well and help them be more innovative.

With these two ways of thinking embedded in the culture of your classroom, you can begin to nurture students who can think creatively and find innovative solutions to complex problems. What other ways do you use math instruction to promote innovation and reasoning? How do you get your students to think outside the box? Share your ideas in the comments below.

Want to learn more? You can explore Conjecture and Chaos more deeply, as well as three more habits of an innovative mind, Communication, Collaboration, and Celebration, in my new book, *5 Principles of the Modern Mathematics Classroom*.

References

Medina, J. (2014). *Brain rules: 12 principles for surviving and thriving at work, home and school *(2nd ed.). Seattle, WA: Pear Press.