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?

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.

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?

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.

What math class looked like today.

Then we compiled the results.

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

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.

Composite Figures in Context: The Wedding Cake Problem

What you see below is the original post from 2013. Since that time, this problem has taken on some alternate forms. One alternative was suggested by Jeff in Michigan and appeared in a post in March of 2014. The other alteration I made myself in Desmos in May of 2015. Feel free to read on and consider checking out the many different ways there are to approach this interesting (and delicious) mathematical situation.

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.

taken from Geometry, copyright Holt, Rhinehart, Winston 2007, Pg 684 #8

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

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.

This blog does not endorse Betty Crocker, General Mills, or any of their products.

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...

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

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.

What if this is actually true…?

Last August, Kelsey Sheehy (@KelseyLSheehy) published “Failing a High School Algebra Class ‘isn’t the End of the World’  in US News and World Report. In the article she discusses the arguments of a couple of critics who are essentially arguing that it is time to reassess the long-held tradition that algebra is an mandatory part of a high school mathematics curriculum.

From the article:

“Algebra requirements trip up otherwise talented students and are the academic instigators behind the nation’s high school and college dropout rates, argues Andrew Hacker, an emeritus professor at CUNY–Queens College and author of the much-debated article.” (Sheehy, 2012)

She continues:

“Hacker argues that students should understand basic arithmetic, but memorizing complex mathematic formulas bring little value to society. “There is no evidence that being able to prove (x ² + y ²) ² = (x ² – y ²) ² + (2xy) ² leads to more credible political opinions or social analylsis,” Hacker writes.”

Sheehy is referencing Mr. Hacker’s Op-Ed from the NY Times dated July 28, 2012, which has generated a rather strong response. The comments number 477 and are closed. Some of them are less than cordial. There have also been blog responses like this (which also got 20 comments).

The last paragraph of his piece makes no bones about his point-of-view:

“Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions. Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven’t found a compelling answer.” (Hacker, 2012)

It would seem that the policy-makers across the country would disagree with him, but I’m not sure our policy-makers are any more equipped to make that decision than Mr. Hacker.

I’m also considering Dan Meyer’s (@ddmeyer) rather engaging line: “I’m a high school algebra teacher. I sell a product to a market that doesn’t want it, but is forced by law to buy it.”

So what gives? If Algebra is driving up failure rates, driving down student motivation, and, as Mr. Hacker asserts, isn’t actually useful (a very debatable point but perhaps a debate worth having), why do we continue to insist that Algebra continue to be mandatory?

And if it is going to continue to be mandatory (which might be the right thing to do), what can we do to make it more universally accessible?

Perhaps the question of why algebra is mandatory should give way to the much more meaningful discussion: Why do kids hate it so much and how do we fix that problem?

Show me the money… or something else useful…

It's not all about the money... photo credit: Flickr user "401(k) 2013" - used under Creative Commons

It’s not all about the money…
photo credit: Flickr user “401(k) 2013” – used under Creative Commons

In Sir Ken Robinson’s (@SirKenRobinson) book Out of Our Minds, he describes an economic model for our education system that is grounded in Enlightenment era philosophy.

According to Robinson, The Enlightenment is responsible for the labeling of topics as “academic.” At the risk of oversimplifying it, things that can be empirically supported are academic and things that cannot are non-academic.

For example, imagine a sunny day. According to our Enlightenment-conditioned minds, we could talk about “academic” things like the convection caused by the warming earth, the refraction causing the sky to appear blue, the air pressure causing the gentle breeze or the photosynthesis making the grass grow.

We could also talk about a lot of supposedly “non-academic” things like how beautiful the deep blue of the sky is, the lift in our spirits that comes from the sunshine, or the memories of when we were kids in the summertime. (Of course, we could try to make these academic by talking about the sunshine releasing hormones that effect the brain which lifts our spirits, or something like that.)

We’ve also labeled people as academic and non-academic. You see, anyone can feel the warmth of a sunny day, but only the smart, academic kids can understand and discuss heat transfers due to radiation from the sun, right?

And those are the smart kids who do well in math class. And those are the smart kids who get good jobs. By good jobs, we mean jobs that pay a lot of money. And if, you can make yourself academic, you can get a good job that pays a lot of money. You’ll be a smart person, too!

This message has created websites like this or this .The message: The good jobs need smart people. Math is the key to being (or looking) smart. Be smart and get paid well for it.

This message has understandably fostered a response in websites like this, which exist to assure kids that they are able to make money without the mathematics.

But wait, wait… WAIT! Why are we connecting math class to money? Does my “useless” math class only exist to get people high-paying jobs? Surely their must be a REAL reason that my classroom is full five times every day. What about people who don’t want one of those smart, mathy jobs that pay well? Equating math to money excludes significant chunks of students. It excludes future homemakers, military personnel, farmers, people who intend to follow into the family business, or people whose future goals include jobs that they KNOW aren’t going to pay well (teachers, artists, musicians, trade laborers, to name a few). To these folks, a math class that exists to get them paid well truly is useless.

Have we convinced these people they’re dumb because my math class is useless on those terms?

The worst part is that “useless” math classes (like the ones that I teach) are actually useful to all of those people. Math is more than a future paycheck. It is more than getting labeled smart or dumb. It is more than a key to some future door that you won’t appreciate now, but will be so thankful for later.

Maybe my “useless” math class can be for them. All of them. To use right now. To learn how to solve problems. To develop a linear sense of logic. To practice the art of questioning, of guessing well, and of learning to check an answer. To increase numeracy. To learn to struggle and to be patient. If my math class can do these things, then maybe my “useless” math class isn’t actually so useless after all.

The Pay Scale Problem


photo courtesy of flikr user “buddawiggi” – used under creative commons


So, we recently earned a new contract at my workplace and it got me thinking about how pay scales work in education with the steps and the percentage increases and all sorts of stuff.

So, let’s see what we can do with this one.


photo courtesy flickr user “Victor1558” – used under creative commons

Let’s suppose that you sign a contract in which you are entering the work on step zero (education-speak for “entry level”).

The pay scale starts at $30,000 and each step increases $2000. Each year you work, you go up one step.

However, also written into the contract is a 1% raise each year that you work. So, the whole pay scale will increase 1% at each step each year.

How many years will it take for an entry-level teacher to begin a year earning a salary of $50,000?