What’s the Opposite of Success?

I want to explore two very important questions: How? and Why?

These questions tend are at the heart of the talk over how to “reform” America’s schools. Everyone from Sir Ken Robinson and Dan Carlin to Sal Khan and President Obama has an ideas. Some I agree with (the first two), some I’m not completely sold on (the last two). Either way, they all agree on this: something’s wrong and we need to fix it.

How do we go about fixing things? Bush’s No Child Left Behind, with it’s younger brother Race to the Top has something of a logical plan for creating success. Except that is isn’t really working very well. When it doesn’t work, then you have to move past the why-it-should-work-explanations and move toward a new set of discoveries.

What if there was a place where education is working well and we could explore HOW the system works. We can deal with WHY later. We have theorists and researchers who love to publish papers. They can work on the WHY. I am a practitioner. I am entertained and engaged by the WHY question, but find the answer to the HOW question more fulfilling.

One place has found success by doing the following things:

• Narrative grades only until Grade 5
• After that, teachers stress grade “as little as possible”
• Not comparing schools or students by standardized testing
• Teacher training programs resulting in each teacher having a master’s degree with the expectation that they will be “experts of their own work.”
• Allow high levels of teacher autonomy by not mandating curriculum from the top down
• Highly emphasizing “soft skills” like analysis, creativity, collaboration, and communication.

By doing this, they’ve created a national school system that is among the very best in the world.

That place: Finland.

Now, before I get labeled a “Finn-o-phile,” I want to state up front that I have no particular affinity for the Finnish culture (although, full-disclosure, I am partially of Finnish descent). I am focusing on Finland’s system because it is working better than ours.

To explain, I am going to let two articles most of the heavy lifting on this one. One from the Globe and Mail out of Toronto and the other from The Huffington Post.

I want to isolate some quotes from these articles:

The First from the Globe and Mail:

One of the ways the Finnish education system accomplishes [its success] is by giving individual teachers greater autonomy in teaching to the needs of their classes, rather than a top-down, test-based system.

America is currently moving away from this model. You can think good things or bad things about the content of Common Core, but the message is clear: Across the country, we want everyone doing the same thing. The Finnish system does have a National Core Curriculum which are defined  as “the legal norm for educational institutions” (sincere thanks to Dan Meyer (@ddmeyer) for fact-checking me on that) although discussions of assessment are much different than those of NCLB(which mandate statewide testing) and instead focus on assements “guiding and motivating” students as well as developing “their abilities in self-assessment.” (Quotes from the Finnish National Board of Education)

Also from the Globe and Mail:

The reality in Canada, which is unfortunate in Dr. Sahlberg’s view, is that students are rewarded for competing against their peers, teachers are held accountable by their class’s performance on exams, and schools are compared through widely published standardized test results. Finland takes an alternative approach.

The story is the same below the Canadian border as well. Standardized testing is THE evaluation tool for most schools and teachers. Real estate agents love it because it is so ingrained in our culture that parents will move into communities with good test scores because we’ve been conditioned to think that those are measures that tell the whole story. The Finnish system does the opposite.

Also from the Globe and Mail:

In addition to emphasizing collaborative work, Finnish schools have a different conception of knowledge than the traditional one. Teachers don’t think of knowledge as a cumulative store of objective information. “It is not primarily what individuals know or do not know, but more what are their skills in acquiring, utilizing, diffusing and creating knowledge that are important for economic progress and social change.”

Perhaps exposing a bit why standardized testing is avoided in Finland, these “soft” skills that are difficult to assess off a bubble sheet. According to Finnish National Board of Education, the National Core Curriculum includes options for on-the-job training with flexible assessments in which student can earn credits through “set of work assignments, a written paper, report, project assignment, product or equivalent” completed “performed individually, in a group or as a more extensive project.” American policy-makers are starting to appreciate these skills. Indeed, have you read about the Smarter Balanced Assessment? Leave it to us Americans to try to find a way to create such a standardized test.

From the Huffington Post (quoting Finland’s Minister of Education, Ms. Henna Virkkunen)

Our students spend less time in class than students in other OECD countries. We don’t think it helps students learn if they spend seven hours per day at school because they also need time for hobbies…

We seem to think that if students are struggling, they need more time in school. The Finnish system does the opposite.

So, let’s recap: Less time in school. Less testing. Less competition. More success. Could you imagine an American Politician standing on that platform?

The Finns have produced a system based on trust. They trust the teachers, they trust the local districts, they trust the students. The American system is based on a lack of trust. We call it accountability. We mandate curriculum because we don’t trust local districts. We over-rely on standardized tests because we don’t trust the teachers. We want longer school days because we don’t trust the students.

There is a nation that is excelling at education. They are, in many ways, doing the exact opposite of the things that we are doing. We, who are eagerly seeking to improve our system, are putting our hopes in standardized testing and state and federal manipulation of school districts through funding incentives. Perhaps it’s too early to state boldly that American reform efforts will fail, but we can say boldly that there are places where real excellence is happening and those people are moving in the opposite direction.

We could spend weeks arguing/discussing/explaining about WHY the Finnish system works. Don’t get me wrong, that is important. But, what matters most to me is this: It works. We could be doing what they do. We’re not… and it appears we won’t be for the foreseeable future.

Double Stuf Oreos: Are they really double the Stuf?

Double Stuf: A factual statement or clever marketing trick?

I was inspired by this post by Nathan Kraft (@nathankraft1) in which he engages his staff in a question about Oreo cookies. (Mr. Kraft was quick to inform me that Christopher Danielson (@Trianglemancsd) was the inspiration for his post. I do want to give credit where it’s due.)

I decided to see what my third hour students would do with it. So, last week Friday I showed them the picture and we started discussing the a variety of aspects of Oreos (some of which were more useful than others). Then, it happened. One students asked:

“Is the stuff of a Double Stuf really double of the stuff in a single stuff?”

The beauty of this activity is that the students were able to become involved in the formation of the solution process. They practically all had a prediction. First idea, would double the stuff be twice as tall?

It didn’t appear to be double by height.

The above image represents what multiple students observed. It was an awesome opportunity to discuss conclusions. What conclusion can we draw from the observation we just made?

Either it was double the stuff and it wasn’t manifesting itself in its height, or it wasn’t double the stuf. (Often, the student’s original predictions colored their conclusion to these observation.)

Double by what measure? Mass?

Next idea was mass. Gave in impromptu call to Mr. Corcoran, the chemistry teacher, who loaned us some scales. But what do we measure? The whole cookie? That opened up another important question? Is the same wafer used for both the standard and the double-stuff?

After some quick diameter and mass measurements, it seemed like there was no meaningful difference between the two. But, just to be safe, each student scraped the Stuf from a standard and a Double Stuf and set to the scale to get a mass measurement.

What math class looked like today.

Then we compiled the results.

The mass of the Stuf scraped off a sample of standard and Double Stuf Oreos.

Each group took a moment to deliberate and concluded that, for the most part, it seems that the Double Stuf is appropriately named. Some groups seemed to think that, if anything, the Double Stuf contained more than double the Stuf.

This activity contained so much of what makes contextual, collaborative learning valuable. Authenticity, source of error, conclusions that were not clear, but needed to be discussed. Students needed to listen, speak and rephrase when others didn’t understand.

It also had the beautiful feature of me not knowing the answer and they knew it. So, there wasn’t the temptation to treat me like the math authority, as though all math learning begins and ends with the Teacher’s Edition.

And for an added bonus, the AP Stats class meets next door at the same time and so, we were able to strike a deal to rerun the trial with the guidance of the stats class for a broader, students-teaching-students experience.

I’ll report back with our findings.

Isn’t Geometry an Art?

The things I do to get the students to learn geometry…

By the time most students arrive in my classroom to take geometry, they have seen Math 7 and Math 8 (which are two sides of the same pre-Algebra coin) and Algebra I. The stated goals of this sequencing is to “prepare the students with the necessary algebra skills to be successful in Algebra I (and later on in Algebra II) and also to be successful on the state tests (currently the Michigan Merit Exam, soon to be replaced by the Smarter Balanced Assessment).

And then there’s geometry…

And our textbook decides that it wants to teach this class right in line with the sequencing pattern: right between Algebra I and Algebra II comes “Algebra G” or Geometric Algebra. I’ll admit it makes some sense. We have spent several years training our students to “do algebra.” Typically that includes plotting points, graphing lines, solving equations and systems, and manipulating functions. If that has been the environment for the last three math classes, then, for the sake of the students, it would make sense to keep the model the same, right? Consistency and predictability breed success, right?

But what’s the risk? Well, as I’ve talked about before on this blog, the Algebra taught in most places is significantly lacking in it’s ability to engage students. So, while the theory is that our students come in well-prepared and well-trained, the reality is that most students come in with sensitive pressure points and calluses similar to someone who has walked for three years in the same pair of shoes. While the shoes are certainly familiar, they might also be smelly and worn out. The ankle support might be gone and the students have learned how to walk funny in them to avoid the blisters and shin splints that have plagued them in the past.

Is this any less geometry then having the student solve equations?

Perhaps what some of them need is a new pair of shoes.

Today’s learning targets is getting students to be able to visualize cross-sections and 3D shapes made when 2D shapes are rotated around specific lines.

To approach that, I started with this activity and I noticed very quickly that the students started “walking funny.” I had asked them to put on their old Algebra shoes and the predictable disengagement started setting in very quickly. I wasn’t going well. Following the Dan Meyer Model, I could find at least three indicators that I was doing math education wrong.

Then something dawned on me: Why do we insist that students do algebra all the time? Geometry is the measuring of the earth, a true down-to-earth, visual math. Why can’t it be art?

So, we changed things up. I had them all get a piece of paper and I acted like an art teacher. I put stuff on a stool and had them draw what they saw.

Engagement hit 100% fairly quickly, especially once I struck a pose and asked them to draw me (see the pic at the top).

If 100% are willing to do this, and 40% are willing to graph lines, which is a better math activity?

Apparently, the new shoes I gave them to wear were a lot more comfortable and many more of them were willing to follow along the path. Content-based discussion and collaboration began to happen without me telling them to do so. As a result, when we had to discuss how the activity supported the learning targets, the connections were much better.

Then I gave them back the original handout (which, for the record, I still think is a good handout), and the students did much better with it the second time.

So, I ask again: why do we insist on turning Geometry into an Algebra course? It doesn’t seem like it has to be. Our current textbook says it has to be, but the textbook only has the authority we choose to give it. (See The Blessed Textbook Conflict if you want my take on the authority of textbooks.)

Today’s activity would suggest that if we are willing to consider what a class would look like if we really wanted the students to simply learn geometry, perhaps it would look more like an art class instead of a math class.

Composite Figures in Context: The Wedding Cake Problem

We’ve started our 3-D unit.

Once we get into the volume and surface area measures for 3-D figures, the textbook leads us to shapes called “composite figures” that look like this.

source: Geometry, copyright Holt, Rhinehart, Winston, 2007, Pg 684 #8

This can be a tricky image for student to try to work with, mostly because they’ve never seen anything that looks like that before. But they’ve seen composite figures. They are everywhere. But, removing the context can be enough to take this very applicable, contextual concept and make it abstract enough to be confusing.

In reality, composite figures are wonderfully applicable. (I say again, they are everywhere!) So, here’s my question: Why do we insist on giving them an abstract picture to start with? Why not start them with one of the many composite figures that will draw the students into a real context.

I present exhibit A: The Wedding Cake.

Photo Credit: Flickr user “kimberlykv” – used under Creative Commons

Your basic wedding cake, like the one shown at the top is three cylinders of differing sizes stacked on top of each other. What I like about the Wedding Cake is that the measures of volume and surface area matter in real time and without too much more background than a bit of story-telling (which I love to do).

Now, let’s toss an additional cylinder into the mix.

Now, you’ve got yourself a math problem!

Question to the students: How much can the baker of the above cake expect to spend on the lemon frosting that is on the exterior of that cake?

… and see where they go with it.

Cereal and Peanut Butter: The Unexpected Lesson Plan

The unplanned econmonics lesson helped it all make sense…

We’ve spent the last few days picking apart The Ritz Cracker Problem, Episode I. I designed this problem about two years ago and this is the first time I have unleashed it onto a group of students. I wasn’t sure what to expect. I set my learning goals and after some individual deliberations, we started big group conversation with the question that you see below.

The learning goals had to do with volume and surface area

Translation: If you stack 16 crackers up and then split them into two stacks of eight, can we simply avoid using the volume and surface area formulas by simply dividing the values for the 16-cracker stack in half?

As the discussion continued on this point, it became clear that you could divide the volume in two, but the same wouldn’t be accurate for surface area. The explanation for this became a bit of a sticking point for some.

Then peanut butter and cereal come to the rescue. (I teach in Battle Creek, MI. Cereal is involved in everything we do, after all.) I never even thought of this image. It never crossed my mind.

I asked them why a 16 oz. container of peanut butter or cereal could be, perhaps, $2 but double the amount would almost certainly be less than $4. How is that manageable for the company selling the product?

Now, that is fairly complex answer in reality, but for our purposes in class, the students were able to explain and understand that bigger packages allow the company to push way more product for a minimal increase in packaging. Bigger cereal boxes allow Kellogg’s to sell more of what they make (cereal) while not having to spend time and money messing around with what they don’t make (boxes).

Translation: Combining smaller packages allows can allow for big changes in volume without a correspondingly big change in surface area. The peanut butter and cereal did it!

… and I never saw it coming.

I Hope You Struggle Well

How prepared are our students to struggle well?

One year while I was saying my final goodbyes toward the end of a final class period of the school year, I scandalized my students quite effectively. It had been one of the most enjoyable classes that I’d ever worked with and I told them so. That part didn’t scandalize them. What did was my justification for why I enjoyed them so much.

” All year long, you all struggled well. You struggled together.”

They looked surprised. Some looked indignant, as though I was being sarcastic with my original statement. But that was sincere praise and it is some of the highest praise that I will give to a math student. What I meant was that they had been a resilient group who understood the learning process with its ups and down. They accepted that it wasn’t going to always be easy and worked through it. Sometimes they struggled and they struggled well.

We need to encourage the ability to struggle well.

There is a logical conflict that exists in education. We want kids to succeed (meaning get good grades… meaning get a lot of answers correct… meaning be able to do what a teacher asks at least 85% of the time.) We also want rigor. (Meaning things that are difficult… meaning tasks that students are NOT able to do… at least, not at first.)

Between the initial rigorous task and the final grade-gathering answer there should be a fairly substantial amount of time. Now, what exists in that time between the introduction of the rigorous task and the submission of the final answer depends largely on the nature of the instructor, the nature of the task, and the nature of the student.

What should happen is that during that time the students are learning how to struggle well.

Not all attempts end well. That’s okay.

But, depending on the nature of the instructor, the task, and the student, struggling well is not an inevitability.

I know the anxiety of watching the students struggle. The students struggle because they don’t know what to do. It’s the instructor’s job to make sure they know what to do, right? If they don’t know what to do, there is a temptation to believe that I am not doing my job well.

Additionally, some tasks are more conducive to struggling well than others. The task needs to be engaging. The task needs to inspire collaboration. The task needs to have several steps. It helps for the task needs to have an answer with enough uncertainty that testing and reasoning about the answer is necessary.

Many students don’t want to struggle. They want to know answers… now. They want their work to be easy and are willing to object when it isn’t. Beyond that, though, most students have to learn how to struggle well (which is a little bit of a bizarre statement to some). They need to learn how to be patient, how to ask questions, how to test their own thought-processes, how to explain their thoughts to others and listen to other explanations, how to be creative and how to participate in creative processes with others. These aren’t innate qualities in many young people, but if young people can learn these skills, the possibility for intense, meaningful and lasting math understanding is real.

So, because of the incredible upside, I hope our students receive rigorous tasks. I hope they need to be patient. I hope they need to collaborate. I hope they need to reason about their answers with others. I hope they learn to be comfortable being a little uncomfortable. I hope they recognize learning as a process.

I hope they are learning to struggle well.

Risks and Rewards of Partial Credit

photo credit: Flickr user “tehbieber” – used under Creative Commons

Let me tell you what’s been on my mind lately. Partial credit. By partial credit, I mean assigning a point value to a student response that is less than the highest possible point value for that problem because the student didn’t get the final answer correct, but the solution process wasn’t completely incorrect. I have traditionally used this as a grading technique for for multi-step tasks.

In some ways it seems natural. You are giving a student points for evidence of learning. If a student is attempting a 3-point task, fumbles up on one single step and so ends with a wrong answer, shouldn’t the student get to salvage a bit of “credit” for performing portions of the problem correct?

In some other ways, though, it would seem natural that if a student gets incorrect final answers for most or all of the problems on a math assignment, the score wouldn’t be very good. However, through the use of partial credit grading, it is possible for a student to earning a decent grade on an assignment having not actually gotten any of the questions correct.

This problem has blown up on the last unit test for my Algebra II students. This is a class where I am one of three people involved in teaching and planning and I have been out-voted to create exclusively multiple-choice tests, which, becomes a serious problem if students aren’t getting final answers correct. The grades that went into the book were stress-inducing for many of my students.

So, therein lies my conflict. I hope that blogosphere will be willing to chime in to help me.

It could be that I’m too liberal with my use of partial credit. (It has occurred to me to award no more than half-points to a student response that has an incorrect final answer, not sure how I feel about it.)

It could also be that I’m worrying for nothing. (It has occurred to me that if a student can earn 80% of the points on an assignment, perhaps they, in fact, know 80% of the math and so it is an appropriate grade. The 20% they are lacking simply consists of parts that mess up the final answer.)

It could be that this is a problem only because of the difference in assessment styles. I am not a big multiple-choice fan and so I often give constructed response questions.

A quick editorial statement: This all seems to be an issue because our system seems addicted to assigning grades to everything. My favorite solution to this problem would be to eliminate the assigning of point values and grades, but that seems like a long shot at this point…

Thanks for your help everyone.