What I want to express in this blog post is not anything new or innovative. It is nothing that hasn’t been said before.  However, it is something that’s been mulling about in my brain while I was drinking my morning coffee and watching the Twitter stream from the Discovery Educators Network Leadership Council Symposium.

A video kept getting re-tweeted in the stream so I figured I’d better check it out.

You can watch the 2 minute video, Microsoft Labs 2019 Vision:

www.youtube.com/watch?v=DQdGvfV4WnU

As soon as it started I felt like I was watching a car commercial. It was flashy, well-produced and fast-paced. I honestly was not that impressed. I guess what people felt was that it was a window into what the future holds for technology and digital devices.

That I won’t deny.

The name on the video is “Microsoft Office Labs 2019 Vision Montage.” This is the vision that Microsoft has for our future.

What’s wrong with this picture?

Many things.

For one, why are we letting Microsoft dictate what the future of digital life will look like? We could make the same statement about Apple or Sony or any other companies who manufacture digital products.  Many of these companies do use customer input and feedback to improve their products, but in reality we are all consumers of what these companies feed us.

What does this mean for education? It means that we need to be putting our students to the task of deciding what THEY want their future to look like. We live in a time unlike any other in history. Our natural resources are disappearing, we have devices that are more powerful than ever before and we have tools that allow us to connect with people thousands of miles away in a matter of seconds.

Companies like Microsoft are not in the business of planning for the future of our children as members of society or for the future of our global community. We must empower our students with that charge. It is they who will inhabit the future. We must also ensure that we empower ALL students to take part in the building of future society, not just the ones who are privileged and can afford it.

There are many obstacles to overcome when we begin to ask our students to solve real world problems. Solutions to real world problems don’t fit on a standardized test. Solutions to real world problems take time to understand and even more time to solve. Solutions to real world problems require a restructuring of school as we know it.

I have been having various conversations (and sometimes debates) about what it means to be a teacher and a learner in the 21st Century. Some of the conversation has been focused around guiding students to understanding rather than delivering content, creating learning environments where learning is a connected and social experience, and infusing technology into learning when it can transform the learning experience.  The world our students will inhabit will require them to collaborate with peers, understand social media tools and be problem solvers within their own communities and the larger world.  We need to prepare them for that world.

Schools need to allow for tinkering. Tinkering with ideas, tinkering with materials, tinkering with students’ perceived limitations. Tinkering teaches children how to learn from failure. Tinkering teaches children how to think about a problem or a project from many perspectives. Tinkering allows children to build self esteem and feel pride in what they do. Students who tinker are the students who build our future.

Some examples of what I’m talking about:

• The Tinkering School teaches children how to build and guide their own learning. While it is not a true ‘school’ it is a model that could be replicated on a smaller scale within the school curriculum. (Listen to the founder’s TED talk here.)
• Philadelphia High School Students Design the Car of the Future
• Project Based Learning motivates students to solve real world problems–Edutopia article
• Sylvia Martinez writes extensively about tinkering on her blog, GenYES

There are those who will look at these words as a ‘pipe dream,’ ‘utopia’ or ‘fairytale.’  To them I would argue that we must have a Vision. If Microsoft can construct a vision of what it thinks the world will look like in 2019 then we as educators, parents, community members, lawmakers and general stakeholders in the world need to have a vision, too. Even more importantly, we need to let our children begin to build their own vision for their own future and give them skills to make it real.

How would you sort these?

Take a look at this picture. If I asked you to sort them into piles, how would you do it? OK, now do it again a different way. No problem, right? Again. Took a little longer for you to think of a way to sort them this time, didn’t it?

I’ve done this with kids and adults of various ages. The first few times we sort, it’s simple and straightforward. The next few times it starts to get more challenging. Eventually there are people sitting there thinking, “There is no other way to sort these!”

When people have gotten to this point, I’ve said something along the lines of, “You have to stretch your thinking. Be creative!” This would often just result in frustration for both of us.

Now I know why. According to this article in Newsweek, telling someone to “be creative” can actually have the opposite effect, closing off their thinking and making it more rigid.

So how can we help our students become more creative? Try some of these strategies:

• Plant the seed. Instead of a vague “be creative,” tell someone, “give me an idea that only you could come up with.” According to Marc Runco of the University of Georgia, this simple switch in directions can double the student’s creative output.
• Make it messy. Creativity is squashed when students feel like they are looking for one right answer. Give students problems that have multiple solutions. Even better, give them problems with no clear solution. Mucking around in the problem solving process can free up creative thinking.
• Never accept the first answer. Even if a student gives you the response you were expecting, say “Can anyone think of another answer?” or “Is there another way to do that?” It sets an expectation that one answer, even if it works, isn’t the end of the process but just the beginning.
• Teach creativity techniques. We often think of creativity as some sort of ethereal aura that some people have and some people don’t. In fact creativity is a skill and a process. It takes work and it can be taught. Techniques like SCAMPER can give kids a concrete handle on something that can seem abstract and complicated.
• Reverse the roles. Instead of giving an assignment to students, ask them to tell you what they would do if they were the teacher. “What would you ask the class to do to show they understood this unit?” Share the best ideas with the class and let them pick their assignment.
• Get out. Changing the perspective can change students’ thinking. Hold a class in the cafeteria, or the auditorium, or the football stadium. Or in a living room, on the sidewalk, or in an amusement park. Rearrange your classroom or your schedule.

And before you think, “That’s not possible in my school,” take a minute and come up with a way to make it happen that only you could think of. Or ask your students to figure it out. You might be surprised at what they think of.

So what did I miss? What are your surefire methods for getting your students to think and work creatively?

When I first moved to Bucks County, I knew the major routes to get around the area. I could, by rote, drive from my house to my in-laws’ house. I could also drive from my house to the school where I worked. I could flawlessly and efficiently travel those well-worn paths and arrive promptly at my destination.

One day, I received a simple phone call from my wife: “My parents are making dinner for us tonight. Just come straight from school and meet us there.”

Not a problem. I left work at my usual four o’clock and with traffic arrived a little after 5:30 PM.

“What took you so long? Did you have a meeting after school?”

“No, I left as soon as I could.”

“But it should only take a half hour.”

“That’s impossible. It’s more than that just to our house, then another 40 minutes to your parents.”

“Um, no, dear. There’s a more direct route.”

Turns out I had driven a half hour south only to turn around and drive a nearly parallel route back north to their house. If I’d just gone east instead, I wouldn’t have had to sit through that light four times.

The problem wasn’t that I didn’t know how to get there. I didn’t get lost, I didn’t get confused. I did what I knew how to do. The problem was that I only knew a very specific path and had no idea how the various routes related to each other or where my destination was related to my starting point.

Learning the route is easy. Learning the whole map is hard.

It is a great temptation in teaching to teach students the route instead of the map. It’s faster, simpler, and more often than not produces the right results.

We can’t give in to that temptation, though. I recently taught a lesson about estimation to a group of fifth grade students. They had memorized a multi-step procedure for transforming a number into its rounded version. I quickly discovered, though, that like the students could do little more than mindlessly play back the recording of the algorithm. Many of them got the steps confused, or missed some, and since they had no idea how the process fit into the greater picture of what they were trying to accomplish, they didn’t recognize that there was a problem. When I asked them to explain what rounding was for, for the most part, their answers were along the lines of, “To get a rounded number.” Several committed the common error when asked to estimate a sum of adding the two original numbers then rounding the answer. Most used the words “rounding ” and “estimating” interchangeably.

All of this could have been avoided if the teachers in second, third, and fourth grade had taken the time to build an understanding of the function and purpose of estimation, to explain that rounding is just one tool in the estimating toolbox, to build in number sense and develop mental models of what is happening when we round. Before introducing the algorithm.

As I found out the hard way driving to my in-laws’ house, the shortcut is only shorter when it is used in the proper context.

Students come to our classrooms with many assumptions and misconceptions, and it is the teacher’s job to anticipate them, recognize them, and correct them. Here are a few that I have seen or heard about:

• When you add or subtract, always line up the numbers on the right
• When you multiply, the answer is always bigger
• Rockets work because the exhaust pushes against the Earth
• Magnets stick to anything made of metal
• Christopher Columbus was trying to prove the world was round
• The American Revolution was fought over high taxes

Many student misunderstandings are simply a lack of experience. There is a scene in the 1982 movie, Star Trek II: The Wrath of Khan, where Khan, the villain, is trying to hunt down our heroes. Kirk flies the Enterprise into a nebula in order to obscure the ship from Khan’s scanners. After a few minutes, Spock makes an observation about Khan:

SPOCK: Sporadic energy readings port side, aft. Could be an impulse turn.

KIRK: He won’t break off now. He followed me this far. He’ll be back. But from where…?

SPOCK: He’s intelligent, but not experienced. His pattern indicates…two-dimensional thinking…

Kirk looks at him, smiles.

KIRK: All stop.

SULU: All stop, sir.

KIRK: Z-minus ten thousand meters. Stand by photon torpedoes.

Like Khan, our students are intelligent but have limited experience. I wonder, though, how often we reinforce misunderstandings instead of correcting them?

Often in the name of making our lessons accessible or understandable we simplify concepts and use stereotypical examples. Consider geometry, for instance. When we draw shapes, they always look essentially the same:

Standard pattern block shapes

Triangles are always equilateral and point up. Rectangles are always wider than they are long and are parallel to the ground. At the extreme, we even refer to shapes by different names depending on their orientation. I actually heard this statement during a math lesson once:

And if you turn this diamond, it will become a square.

The shape was always a square; the direction it faces doesn’t make any difference.

Try these suggestions to avoid reinforcing the misconceptions of your students:

• Know your own misconceptions. Begin with the assumption that you may have picked up your own wrong ideas in school or from popular media. Review the material ahead of time and look for places where you yourself didn’t quite get it right. (Incidentally, if you read any of the items in my original list and thought, “What’s wrong with that?” you may want to do a little research and find the subtle problems with them.)
• Plan ahead for student misunderstanding. Learn the places where your students are likely to get confused or have preconceived ideas about a topic. Many misconceptions are common and repeated, so it’s easy to prepare for them.
• Use a wide variety of examples. Deliberately choose examples that stretch students’ thinking. Use counterexamples to help them better define concepts in their minds.
• Let students construct their own definitions. By letting students build definitions and explanations around examples you use, you are encouraging them to analyze the examples and understand the concept deeply instead of just memorizing a sentence someone else has provided them. After they attempt to build a student-friendly explanation, you can come in and provide more precise vocabulary where necessary to give them a more concise way to express it.
• Expect students to explain and justify their reasoning. Sometimes students are able to apply a rote algorithm accurately and get a correct answer to a problem without really understanding what they are doing. Asking them to explain, even when their process seems obvious to you, will give you insight into whether their thinking is accurate or has flaws that need to be corrected.

Soon after Kirk changed his tactics to account for Khan’s misconception, he was able to sneak up behind Khan’s ship, ultimately winning the battle. While it is unlikely that the misconceptions our students carry through school will result in such life or death circumstances, we can make our own jobs easier by preventing them in the first place.

SPOCK
                             Sporadic energy readings port side,
                             aft. Could be an impulse turn.

                                           KIRK
                             He won't break off now. If he
                             followed me this far he'll be back.
                             But from where...?

                                           SPOCK
                             He's intelligent, but not experienced.
                             His pattern indicates two dimensional
                             thinking...

                   Kirk looks at him, smiles.

                                           KIRK
                             Mr. Saavik, all stop.

                                           SAAVIK
                             All stop, sir.

                                           KIRK
                             Descend ten thousand meters. Stand
                             by photon torpedoes.

Every year I stand in front of a group of new fourth or fifth grade students and face the most challenging teaching task I’ve ever had: training them to be telepathic.

I always begin with a magic trick. Each student chooses a two-digit number. Then I walk them through a series of simple calculations resulting in a new number. On that page in their math book, they choose a picture and memorize it.

Earlier in the day, a mysterious envelope had arrived in the classroom, marked “DO NOT OPEN…TOP SECRET.” I now open that envelope, revealing a duplicate of the photo they all have memorized. I can read minds!

Of course, it doesn’t take long for the class to realize it was a trick, and I don’t deny it. In fact, I remind them that I began the exercise by telling them I was going to do a magic trick. The point is why I had to do a trick: teacher’s can’t read minds.

“So what does this have to do with math?” they ask me.

“Ah, excellent question,” I reply. “When you put an answer down on a math test or a homework problem, how does your teacher know what you were thinking when you solved it?”

“Uh…she doesn’t?”

“Precisely. But for us to teach you, we need to know how you’re thinking so we can help you learn how to solve problems better. Since we can’t read minds, what’s the only way for us to know what’s going on in your head as you’re solving a math problem?”

If the lesson were outside at night, this question would normally be answered by the sound of crickets chirping. One brave soul usually raises a cautious hand: “Uh…we tell you?”

A simple concept. A difficult task. Actually getting the thoughts from their heads into words—and eventually onto paper—is something that takes much practice and many examples. Yesterday I talked about one of the ways to begin this process by teaching and using the correct vocabulary.

We need to teach students that math is not about rote manipulation of abstract symbols. Those symbols, and the terminology that goes along with them, are tools with two purposes: solving problems, and communicating ideas.

I’ve developed a structure that helps students organize their thinking and chunk the way they communicate it. I tell them, “Wear Your C.A.P.E.”:

The most difficult aspect of this, of course, is the explanation—describing the why, not just the what. In order to help with this, I teach them the Magic Words. Just like using clue words to identify the operation in a word problem (like “all together” signifies addition), these words can help to signify their mathematical reasoning when they talk or write. (This list is based on an article by Diane Hurst published several years ago in the PA Math Assessment Handbook, but no longer appears to be available):

Students who learn to use these words correctly will begin to unpack the reasoning that is going on in their heads.

How could you adapt this to your situation? What other subject areas might it work for? Do you have other ideas about teaching students to be “telepathic” and communicate their thinking to other people?

Teachers of mathematics need to recognize that there is a strong link between language, writing, and problem solving. In most of the assessments that states use to determine student and school success, a student must demonstrate math reasoning abilities through writing. This skill is not automatic, though. It develops through a recursive process:

Vocabulary & Language <—> Reasoning <—> Talk <—> Writing

Beginning with vocabulary and language, a student learns to reason, then to communicate those thoughts verbally, and finally to write. Each of the levels feeds back to the previous one, reinforcing and further developing it.

Thus if we’re going to teach reasoning skills effectively, it follows we need to carefully consider the vocabulary we use.

It isn’t uncommon, especially in the primary grades, for teachers to simplify the language we use with children to explain complex concepts. Although this is useful, it can also lead to sloppy language if we aren’t careful. It is particularly important that we don’t permit students to use precise math terms improperly and that we teach the “real” terms as quickly as possible. Even if students don’t use them right away, they should be hearing the correct terminology in context from the beginning.

Here are a few examples of sloppy math language that I often hear from older students. If these go uncorrected, students will have a very difficult time communicating well when they need to explain their thought process–a skill that is essential to upper level math.

I believe it’s essential to require students to be precise when they communicate. Often when students don’t use the correct term, or use a valid term improperly, it is a sign they just don’t have the right words.

I’ve heard teachers argue that young children just aren’t capable of such sophisticated language yet. My father, a retired professor of speech/language pathology, has often said, however, that if second graders can learn and correctly use terms like “Tyrannosaurus Rex” and “Diplodocus”, why on earth can’t we teach them to say “subtracted” instead of “minused”? Vocabulary instruction should be as much an integral part of mathematics as it is of reading, writing, and other content areas.

Tomorrow I will tackle a more challenging vocabulary-related issue in mathematics: verbal and written explanations of a student’s cognitive process.

(This article is based on material I originally posted in Grandé With Room.)

Image by kevindooley via Flickr

Yesterday, I shared some questions that I often use to help create an atmosphere of thinking in my classroom. Unfortunately, when I ask a student to explain their reasoning, they often aren’t able to reflect back on their thought process and verbalize what took place. In some cases, the best they can come up with is “it just popped into my head.”

In order to train students how to do this, I scaffold the process for them at first to give them a structure within which they can build their own responses. They need to learn three skills to allow this to happen:

1. Focus on the process before they start
2. Monitor their reasoning as they are working
3. Reflect back and explain to someone else what they were thinking

Each of these skills needs to be modeled and practiced, and students need many opportunities to use them. These thinking skills are learned best when they are integrated into the regular flow of instruction rather than explicitly taught as discrete topics. One way to do that is to build one or more of these scaffolding activities into every lesson:

• Think-Alouds
• Leveled problems
• Graphic organizers (e.g. T-chart)
• Using “magic words” that students can use which require explanation of reasoning
• Asking prompt questions (such as those in yesterday’s post)
• Give part of the solution, then have students complete it
• Give the answer, students write the solution
• Give the explanation, students write the solution
• Give the solution, students write the explanation
• Checklists or mnemonics to aid recall of processes
• Journals to practice informal writing about problem solving
• Vocabulary games to build language skills and improve communication about reasoning
• Allow students to rewrite weak explanations to improve them
• Show sample student papers that demonstrate good skills
• Teach students to score responses using a rubric
• Have students score their own work or a partner’s work
• Trade papers with another class and have students score

Image by gutter via Flickr

One of the things that I frequently see in classrooms that I visit is students who can mechanically produce an answer to a question or problem but who don’t really understand how or why the process they used works. As teachers, we need to focus more on the thinking process that a student used to get to an answer rather than on the answer itself.

Certainly there are times when simple recall is important, and when it’s best to give students a brief indication of whether their response is correct or incorrect. But for any question that involves reasoning, judgment, assimilation, synthesis, or similar higher level thinking, I like to ask follow-up questions like these:

• “Why did you do that?”
• “How did you get that?”
• “How do you know?”
• “What does that number/fact/word represent?”
• “What does that mean?”
• “Can you justify your answer?”
• “Can you prove it?”

I ask these regardless of whether the initial answer is right or wrong. This has several benefits:

1. I can get a better understanding of both the right and wrong answers a student gives. Was it simply an automatic application of a rote process? Is there valid reasoning going on with simple mistakes? Was the right answer a guess or a fluke? Does the students have a misconception that happens to work right in this instance?
2. Occasionally a student will have a good justification for an alternative answer I hadn’t considered, and asking for the rationale saves me from a hasty dismissal.
3. It makes it clear to the student that they are responsible for their answers, not me.
4. It creates an atmostphere that is simultaneously more rigorous and more open. It becomes safer to be “wrong”, because when they can explain their thinking, we focus on the process instead of the result. It is rare that a student does nothing right in that thinking process, and so we can begin with “I understand where you are coming from. This part was really good thinking, but here is where you got off track and how you can fix it next time.”

So many times I have been in a classroom where a student gives an incorrect answer to a question, the teacher gets a correct answer from another student (or simply provides it him- or herself), and moves on. I’ll sometimes go to that student’s desk and privately ask for the explanation. “Show me how you got that,” I’ll say, and they’ll walk me through the process. It rarely takes me more than a few moments to explain the flaw in the thinking and help the student understand.

Take the time to question everything your students do. Create an environment for thinking in your classroom.