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

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

Unanswerable Questions.

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

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.

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

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.

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

started.

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

Available on Barnes & Noble

 

More from the Hack Learning Podcast

Video produced by Tootsie Roll, 2012

Gerald Aungst Hack Learning Podcast

Gerald Aungst Shares Ways to Make Kids Love Math

Math teachers need to embrace the Principle of Conjecture, according to math teacher/coach Gerald Aungst, author of 5 Principles of the Modern Mathematics Classroom: Creating a Culture of Innovative Thinking (Foreword by Hack Learning creator and podcast host Mark Barnes).

Aungst suggests that no one really hates math–that perhaps they just haven’t been taught in a way that inspires curiosity and conjecture.

5 Principles of the Modern Mathematics Classroom

Look inside

“We want kids to be asking a lot of questions and then seeking those answers,” Aungst says in Episode 53 of the Hack Learning Podcast, embedded above.

Create situations that spark conjecture

Aungst learned how to inspire inquiry and conjecture in kids by creating unique situations where “something doesn’t make sense or is not immediately obvious.”

Once students’ curiosity is heightened, Aungst suggests allowing kids to immerse themselves in the problem. Rather than handing them the correct formula and a worksheet, Aungst says the teacher’s job is to ask questions the lead to collaboration and conjecture. Questions like: “What can you figure out? What can you notice here?”

When this strategy is employed, students often solve the problem with little or no teacher intervention, according to Aungst (learn more at time index 4:38 in the audio embedded above).

What to do tomorrow

The best way to hack math instruction tomorrow, according to Gerald Aungst, is to create a mindset of questioning and problem-solving in students from the beginning of the year and continue it throughout the year. Here are some do-tomorrow strategies to create this mindset:

“How do you know?” This is the simplest thing you can do to raise the level of thinking in a classroom or for your child at home. Ask this every time a kid answers a question.

IWonders: create very open-ended challenges for students that promote lots of inquiry and have multiple paths to a valid solution.

    • Example: Imagine you decided you were going to read every book in your school library. How old would you be when you finished? Such a simple question, but there’s so much going on here.
    • There’s no way to have one “right” answer, so students are going to have to rely on justifying their solutions and choices with solid arguments and evidence.
    • Here’s another example that would work well for older students: when does it make sense to take a longer route in order to have one less traffic light? This might require some research or at least some reasonable assumptions.
    • For both of these, you and your students can decide how precise the answers need to be, and what level of supporting evidence is needed to justify a solution.
    • The same question is easily adapted to different students, too, just by creating different parameters or expectations for the depth of the solution

Find the Problem: Instead of only giving problems and having students work out the solutions, try these variations. All of these are great activities for partner or group work, too.

    • Give the problem and the solution without the intermediate steps. Students have to figure out how to get from A to Z.
    • Give students just the solution steps and answer. They must deduce the problem. For a greater challenge (and greater opportunity for creative and critical thinking), be more vague in your steps. This is also a great way to illustrate why students need to be clear, detailed, and precise in their solutions!
    • Let students start a solution then switch papers periodically. They should then continue the solution from where the previous person left off. You can’t change anything they did, but you can add steps to correct an earlier error.
    • Give students random bits of unconnected information and ask them to generate a problem and solution that somehow connects them.

Aungst says teachers should be asking students questions often about the problem-solving process, in order to create this mindset (listen to his list of key questions at time index 10:30 in the audio embedded above).

Resources

Gerald Aungst is the supervisor of gifted education and elementary mathematics for the School District of Cheltenham Township, in Pennsylvania, and author of 5 Principles of the Modern Mathematics Classroom: Creating a Culture of Innovative Thinking. Prior to his service as an administrator, Gerald taught mathematics at the elementary level for eighteen years in both the regular classroom and as a gifted support specialist. Most recently Gerald has been part of his district administrative team leading the transition to the Pennsylvania Core Standards and training teachers in high quality math instructional practices.

Ask Gerald Aungst a math instruction question in our comment section below or on Twitter @geraldaungst and learn more about his work at his website.
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