The Power of Unanswerable Questions

Let me begin with full disclosure: Before I learned about Unanswerable Questions in Hacking Mathematics, by Denis Sheeran, I thought just about everything math related was unanswerable. Admittedly, math has always been pretty elusive to me.

After reading about Unanswerable Questions, though, I’m looking at math and pedagogy differently.

Rather than talk around Denis Sheeran’s concept, I thought I’d just share it straight from the mathematician’s mouth–or at least from his book.

From Hacking Mathematics: 10 Problems That Need Solving, with permission from Times 10 Publications



You’ve just about finished a chapter in your textbook

or unit in your curriculum materials and then

you notice it, right there, staring you in the face. What

is it you see, menacingly staring back at you? The last

section of the chapter: the statistics section, sometimes

called “statistical connections” or “data around us” or

“modeling.” No matter what it’s called, it gets translated

by a lot of math teachers as “Skip me, you don’t have

time.” But don’t jump on the skipping bandwagon.

Find the mean. Find the median. Find the mode. Make

a bar graph or pie graph. What’s the probability of flipping

heads on a coin? Twice? These are the instructions

and questions that encompass the complete statistical

learning many of us received in middle school, mostly

with data sets of five to ten pieces of information. Some

of us who had more adventurous teachers may have even

made graphs of bivariate data and tried to come up with

our own lines of best fit using completely non-statistical

methods by employing our understanding of writing a

linear equation using two points.





We live in a different world now, where large data sets

are available instantly and calculation tools can organize

and calculate all we need to know in less time than it takes

to sharpen our pencils. It is no longer useful to spend our

time teaching arithmetic and calling it statistics. In today’s

classroom, the mathematics teacher has the opportunity

and responsibility to create statistical thinkers.

Unanswerable Questions will develop statistical thinkers

in your classroom.

Rather than dwell on the past, let’s look at the present

and the future for most of us. Our standards and materials

spell out the statistical concepts we are to teach. What has

changed for our students is that the standards no longer

ask for students to calculate and find statistical values,

but instead to recognize relationships, understand variability

and its effect, and make predictions based on

interpretation of data. In short, true statistical thinking

is missing. Statistics in today’s schools should be based on

Unanswerable Questions.


When we ask students to find the mean of the heights of

the twenty-three students in our class, we are asking them

to average numbers together, which is a very easy question

to answer and an even easier question to grade. Instead,

when we ask, “How tall is the seventh grade?” our students

must begin an investigation that takes them much deeper

into statistics. They will discuss how to obtain the necessary

information, devise a plan (one that likely won’t work

or is completely unrealistic), refine that plan, measure each

other, standardize their measurements, find means, graph

information, and maybe even come across the idea of a distribution

of data. That’s all before the teacher even needs to get involved.

Since up to this point in their mathematical education,

most questions have had numerical and final answers, the

desire to answer an unanswerable question will continue

to motivate the students to work and think and collaborate.

Finally, they will come to a point where they are

satisfied with their inexact solution to the problem, therein

revealing the heart of statistics: using what we know to

infer about what we don’t know until more information

comes along and either changes our minds or gives us a

reason to reopen the question. Unanswerable Questions

will develop statistical thinkers in your classroom.


There’s a reason your textbook or curriculum source

has the stats section where they do. It’s very likely that

it ties into the unit you’re teaching in a deep, meaningful

way. Here’s how to start harnessing the meaning

and inspiring your students to think statistically.

Look at the statistics section first. See what

statistical concepts are connected to the lessons

you’re teaching in this unit, and work

backward. Find an Unanswerable Question

that you can share as you open the chapter,

and refer to the question throughout.

Find claims in the media to discuss. Every

single day, you can find stories in the media

with claims made about a company, a government

office, an auto manufacturer, or a

school. Present students with the opportunity

to debate those claims. It’s likely that in little

time, they’ll need a statistical process to back

up their claims.

Share the unlikely. Lottery winners, survivor

stories, and game show outcomes will foster a

statistical conversation in a hurry. When you

read about them or see statistics in the news,

make note of it and bring it to class to start

those conversations.

Find Unanswerable Questions in sports.

Don’t ask answerable questions, like what

a player’s batting average is now that he’s

struck out three times in a row. Dig deeper

for the Unanswerable Question, like asking if

batting average affects salary in baseball. Or

which baseball stat has the biggest impact on

player salary? Those are tough, if not impossible

to answer.


One of my favorite Unanswerable Questions comes from a

TV commercial that aired during my childhood. It involved

a cow, a fox, a turtle, an owl, and a boy. The Unanswerable

Question: How many licks does it take to get to the center

of a Tootsie Pop?

Show the old commercial to your class—it’s on YouTube.

Then, after fending off questions like, “Why does the

owl eat the lollipop?” and “Is this some kind of fable?”

and “Why isn’t the boy wearing any pants?” you can get


The Answerable Questions:

What are the characteristics of a Tootsie Pop that we

need to take into consideration?

What is a “lick” for the purpose of the experiment?

What needs to be measured, and how?

In sixth grade, students need to be able to recognize that

a statistical question is one that anticipates variability in

the data. While the class is discussing and defining the

components of the Answerable Questions, they will see

that variability exists, even in their definitions, and as such,

will exist in their data. Even when they come to an agreement

on definitions and procedures, they will quickly find

that during the data gathering, different students are following

the procedures differently. This leads them directly

into the next question: What do we do with our data?

Students may have enough mathematical acumen at this

point to be able to make good, if not entirely correct, suggestions

as to what should be done with the data—so let

them. In my experience, by the third or fourth suggestion,

they come up with “Average it all together,” or “List it from

smallest to biggest,” and even “Graph it.” At this point, I

may break the class into teams to complete each of the different

valid suggestions and report back, or I may take one

of the suggestions and run with it, depending on the focus

of our previous and upcoming content instruction.

Click image to learn more

Sixth graders need to be able to describe the distribution

of the data using its overall shape, center, and spread,

and recognize that its center describes all the data at once,

while the spread (variation) describes how all the data is

different from each other. They also need to be able to

display the data on a number line (dotplot or histogram)

and describe the distribution in context.

I expect my sixth graders to be able to say: “After

licking both sides of our own Tootsie Pops until each student

reached the chocolate center, we counted the number

of licks per student on each side. The mean number of

licks was ##. This was more/less than I expected. When

we graphed the data, the distribution was almost symmetrical

except for one point which took many more licks

to get to the center. The median, or middle value, was

less than the mean, and I think that’s because of the large

number of licks it took on one Tootsie Pop. No one licked

more than ## times or less than ## times before reaching

the center.”

Remember, the goal with sixth grade is not to pass the

AP Stats test, but to introduce data-gathering methods,

require correct statistical language, and to develop the

ability to describe sets of data. To extend this to higher

grades, weigh the lollipops first and compare weight and

number of licks as a linear relationship. (There’s a surprise

ending to that one that I won’t divulge). Students

should also discuss whether or not the Tootsie Pops could

be called a “random sample,” and what randomness is

and why it is important.

— end excerpt

Learn more about Unanswerable Questions and other problems that need solving in Hacking Mathematics.

Available on Amazon

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More from the Hack Learning Podcast

Video produced by Tootsie Roll, 2012

Is your lesson relevant?

I get frustrated when I’m asked to do something irrelevant. Don’t you? And yet, many teachers dread when students ask, “When am I ever going to need to know this stuff?” It’s a rather obnoxious way of asking, “Is this lesson relevant?”

Now, picture this. You craft a lesson that is so relevant that you hope some kid inquires! This episode is designed to help you create such a lesson. To help in this mission is an awesome primary source.

Mitchell Charles

Mitchell Charles is an articulate young man destined for academic brilliance.

In World Civilization, we were meandering through a unit on the Industrial Revolution. This topic typically leaves some students cold. My challenge was to make it relevant.

I did this with the help of Elon Musk and Peergrade. Below is the lesson that Mitchell evaluates. Please feel free to commandeer some of it, or all of it!

Applying the Industrial Revolution via Elon Musk

History students often complain that what they study doesn’t seem relevant. You may have heard the cliche, History repeats itself. You may, or may not be persuaded by this idea, but hopefully you’ll concede that the study of history at least gives us templates through which we can better understand the present, making a lot of subject matter relevant.

Technological changes that are at work today have the potential of reshaping the world along the lines of the changes brought about by the Industrial Revolution. Studying how that unfolded has the potential to make a young person more adaptable.

Elon Musk is one of the world’s most interesting and perhaps most impactful citizens. Musk is determined to improve the way you live. He also is determined to help the USA win back it’s manufacturing advantage.

This man has the potential to create products and systems that are as impactful as anything we’ve studied so far. He was born under the yoke of Apartheid in South Africa and as soon as he was able, he migrated to Canada and then the USA. He’s now a US citizen.

I first became interested in Musk when I saw him appear on the Big Bang Theory years ago. 

We’ll focus on 4 of Musk’s objectives:

  1. SpaceX
  2. Tesla
  3. SolarCity
  4. The Boring Company

Job 1: Make it relevant: Become familiar with your topic by reviewing the links and conducting online research.

Job 2: Meet with other students who’ve been assigned the same focus. Dialogue about the company. You certainly don’t have to agree, but consider the views of your classmates.

Job 3: Individually, Respond to the prompts on the appropriate doc on Google Classroom.

Hacking Engagement Again

Click image to peek inside

Job 4: Upload your views to Peergrade.

Job 5: Evaluate your peer’s ideas via Peergrade.

Each student has been assigned a focus. Here are some links, but please don’t limit yourself to only the links I provide.


Start your research by visiting the company website. Then navigate to this objective article. Conduct research on this entity including a video search.


Start your research by visiting the company website. Then, navigate to this objective article. Finally, conduct research on this entity including a video search.


Start your research by visiting the company website. Then, navigate to this objective article. Finally, conduct research on this entity including a video search.

The Boring Company

Start your research by visiting the company website. Then, navigate to this objective article. Finally, conduct research on this entity including a video search.

Elon Musk Reaction Prompt

Build your responses based on your research. Each prompt is worth 10 points. Elaborate on your ideas. Don’t just give 1 word, or 1 phrase answers. Compose a narrative for each prompt.

  • Pretend that you are Elon Musk. You’re appealing to a number of venture capitalists with the goal of having them invest. Provide your audience with a sales pitch. This has nothing to do whether you…as in you personally, not as in you pretending to be Elon Musk…think the venture will be successful.
  • Indicate whether you think this venture is feasible. Don’t just respond yes or no. Explain why, or why not, you think it’ll work.
  • The inventions of the Industrial Revolution changed people’s lives in fundamental ways. Do you believe this venture will change lives and if you do, in what way and if you don’t, why not?
  • Pretend that you’re in this class and it’s 200 years in the future. Your instructor, who is a remarkably improved version of yours truly, asks you to describe Elon Musk, place him in context, and describe his significance.
  • Indicate something that you would like to see invented. This needs to be an innovation that will dramatically improve life on earth. Describe it, explain how it works, and predict how it will improve life.

click image to learn more

Apply the Hack Learning Formula

The Problem:

Students often feel lessons are not relevant.

The Solution:

Craft lessons where kids are challenged to apply what you’re teaching to their lives.

What you can do Tomorrow:

  • Brainstorm real world applications for tomorrow’s lesson
  • Break your students into 4 topics or problems and then challenge them to respond to provocative prompts
  • Have kids submit their work to Peergrade and then watch a virtual Socratic Seminar unfold before your eyes

Teaching relevance is the responsibility of every educator. Teaching relevance will empower you to embrace the question, “Why do we have to learn this?”

Listen to “94-Usher in Relevance Courtesy of Elon Musk and Peergrade…Starring Mitchell Charles” on Spreaker


A version of this first appeared at

kid hates school

My Kid Hates School But You Can Help

One social share can immediately impact change. “My kid hates school,” I shared on Twitter and Facebook, and suddenly I had support from people around the world.

Here’s what happened. I tweeted this then shared it on Facebook.

School must get all kids to comply. School must organize large quantities of children, manage them, sort them, quiet them and control them. For a lot of kids it may not be a big deal. For some, it’s like stealing their soul. — Anonymous

Some weighed in by raging against the machine:

One regular commenter in this Facebook group had this initial response:

No, you can’t. You can advocate, you can argue, and you can encourage him in a variety of ways. But you can’t change the school or the teachers–it’s not in your power. You can help him to find satisfaction in other ways/venues and maybe within the school or within certain classes. Teach him to hack his own way, eh?

After all of these incredible comments and suggestions, I kept coming back to the previous quote.

And I wonder, is this right? Is school change out of my control? Is it out of our control?

Or, is it possible that we have all the power, and we only have to choose to exercise it?

At the very least, one expert, New York Times bestselling author Jessica Lahey, believes that we need to keep showing the evidence of what’s best for kids to educators. Here’s what the author of Gift of Failure tweeted:

What You Can Do Tomorrow

  • Talk about best practices: Create discussions on social channels and at your school about best practices. Ask, “How can we eliminate old school methods and replace them with progressive pedagogy that inspires student engagement?”
  • Share stories like this one: Share podcasts, blog posts, and Facebook statuses, in which parents and educators discuss their experiences about kids who hate school and what they’re doing about it.
  • Encourage participation: If a kid hates school, encourage her to join a club, go out for a sport, or try out for the play. The more kids participate in things related to school, the more likely they are to start enjoying it.
  • Be present: Talk to teachers, school leaders, and parents in your own school district–especially where you kids attend school–about making learning fun. Don’t let them tell you it’s not about the fun; that’s a load of crap and simply not true.

So, what’s my final take on all of this? I’ll continue to advocate for my son, and I’ll continue pushing his teachers–and all educators–to be the absolute best they can be.

With your help, I believe we can make a difference.

More hacky stuff

Find the Facebook discussion here.

Share your thoughts on Twitter at the #HackLearning feed here.

Subscribe to the podcast here.

Can Your Students Spot a Fascist?

Listen to “James Sturtevant Hacking Engagement” on Spreaker.

If you’ve taught a humanities class, you’ve probably recognized how frequently Adolf Hitler comes up. Unfortunately, many kid’s understanding of Hitler and Fascism doesn’t expand much past the Holocaust, and some students can’t spot a Fascist from across the room.

In fairness to kids, though, lots of adults also struggle with this, because, like students, they haven’t been exposed to the characteristics of a Fascist (see list below).

Scott Elliott teaches 9th Grade World History with me. Right before Christmas Break, we were yakking about how we could teach Fascism, our first unit in January, in a more engaging and impactful way.

Scott found a wonderful resource which formed the backbone of the assignment. The article is by Laurence W. Britt and is entitled Fascism Anyone.

The assignment we created challenged kids to rate WWII leaders on the 14 Characteristics Britt articulates and also to apply them to current leaders with authoritarian traits. Here’s a link to the Hyperdoc we posted on Google Classroom.

Britt’s list of 14 Fascist Characteristics

  1. Powerful and continuing expressions of nationalism
  2. Disdain for the importance of human rights
  3. Identification of enemies/scapegoats as a unifying cause
  4. The supremacy of the military/avid militarism
  5. Rampant sexism
  6. A controlled mass media
  7. Obsession with national security. Inevitably, a national security apparatus was under direct control of the ruling elite
  8. Religion and ruling elite tied together
  9. Power of corporations protected
  10. Power of labor suppressed or eliminated
  11. Disdain and suppression of intellectuals and the arts
  12. Obsession with crime and punishment
  13. Rampant cronyism and corruption
  14. Fraudulent elections

This turned into a solid activity in our World History class, but the lesson can be applied outside of the humanities.

Perhaps, there are misunderstandings about important concepts in other subjects. I can certainly think of examples in science.

Applying the Hack Learning model

The Problem

Students freely use words like Fascist, Nazi, and Hitler with limited knowledge of these label’s broader meanings.

The Solution

Expose kids to what Fascism is and then challenge them to apply it, and challenge them to apply agreed-upon standards to a set of contentious circumstances.

What you can do Tomorrow

  • If you teach a humanities course and Fascism is a topic, please steal our lesson and morph it to fit your needs.
  • If you teach a non-humanities class, take a controversial or misunderstood topic, expose students to some agreed-upon standards and then challenge them to apply that knowledge.

Please inspire your kids to pursue objective truth relentlessly. Assignments such as this will nurture this essential disposition!

Why grading and evaluating are not assessment

Hack Learning’s Mark Barnes wrote about revolutionizing assessment by going gradeless in Education Week’s 10 Big Ideas special report. Some readers are intrigued, and some are pushing back.

In Episode 110 of the Hack Learning Podcast embedded above, Mark takes on the pushback and explains why grading and evaluating are not parts of effective assessment.

No, Students Don’t Need Grades, excerpted from Education Week’s 10 Big Ideas article

Technology and social media continue to disrupt education. Classrooms are morphing into maker spaces; STEM labs and media centers are filled with fascinating electronic gadgets. Teachers spend less time in front of the class and more time in the middle of the action. Schools, teachers, leaders, parents, and students across the country are embracing this brave new world.

In the midst of rapidly changing technology, and consequently, pedagogy, there is another fundamental change I would argue more educators need to embrace. It’s a growing movement to alter the one function of education that most stakeholders steadfastly refuse to revise: how we assess learning.

If you’re interested in disrupting education far more than the 3-D printer or smartphone ever could, consider schools and colleges where there are no grades. Imagine classrooms where teachers never place numbers, letters, percentages, or other labels on students’ work; where report cards don’t exist; and where the GPA has gone the way of the dinosaur.

In a gradeless classroom, the perpetual lies that numbers and letters tell about learning would cease to exist. Honor and merit rolls would disappear. There would be no school valedictorian. Clubs that celebrate high performers would disband. Many colleges and universities would change how they admit incoming freshmen, and academic scholarships would need a makeover.

Moreover, teachers would learn how to effectively assess academic performance, and students would become independent learners, driven by curiosity and inspiration rather than by the empty promise of a “good” grade or the threat of a “bad” one.

Now, this may sound like only a big, perhaps even unrealistic, idea. But the gradeless classroom already exists in schools worldwide. While I don’t claim to be the creator of no-grades learning environments, I and thousands of my colleagues across the United States and around the world have turned it into a movement that is helping educators reimagine how they assess learning.

Read the rest of my EdWeek article, part of the 10 Big Ideas in Education special report, here.

Mark Barnes on grades, feedback, and evaluation

  • Grades and feedback are not the same. Grades are numbers, percentages, and letters that label students and mislead stakeholders about what has or has not been learned.
  • There are many factors that impact learning, and teachers must have a conversation with students about those factors.
  • It’s crucial that teachers ask kids questions: Why did you do this? Why this way? What if you had done this instead? What don’t you understand? How can you better understand this?
  • Education needs to eliminate evaluation from assessment. Evaluation is about judging performance, and the only shareholders suited for this type of judgement are the learners. Teachers should relegate their feedback to observations and questions and teach kids how to judge their work.

More on going gradeless

Click image to look inside Hacking Assessment

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