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
THE PROBLEM: STATISTICS FILL DAILY
LIFE, BUT NOT MATH CLASSROOMS
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
THE HACK: ASK 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
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.
WHAT YOU CAN DO TOMORROW
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
• 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
THE HACK IN ACTION
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.
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
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.
More from the Hack Learning Podcast
Video produced by Tootsie Roll, 2012