Using Imagination in the Math Classroom

What does your mathematics classroom look like? Does it
look like the math classroom you experienced through childhood?
Were you bored? Are your students bored? Are they engaged in the
material presented? Do they enjoy learning? Do they look forward
to your class? Do they participate in mathematical thinking and
inquiry? Simply put, do they like mathematics? Reflect on these
questions while you transport yourself into one of my recent tenth
grade math classes.

I called the class to attention and began. “Once upon a time
there was a little boy named Carl who lived in Brunswick, Germany.
The year was 1785. Little Carl was a very smart boy even
though he came from a poor family. He taught himself to read and
write—as well as to add, subtract, multiply, and divide —by the
age of three. At the age of seven, he attended school in a one-room
schoolhouse with students both older and younger than himself.
Even though he had older classmates, he was the smartest in the
class. The problem was that he was also the poorest behaved.

You see, Carl learned very little at school because he was
already so smart. Some even say that he knew more that the teacher.
I paused slightly, knowing what was coming.

“It sounds like me, Mr. Wilke,” quipped Tim.

“Funny guy, Tim. Funny guy,” I responded laughing with the
whole class. I continued, “Anyway, because Carl was so bored, he
did things that made the teacher very angry. He was also the first to
answer questions the teacher posed to the class. This annoyed the
teacher even more. On one particular occasion during a math class,
where Carl seemed to be crawling on the ceiling while answering
every question, the teacher could not take it anymore. Finally she
said, “Carl I am sick and tired of you. Go into the corner. I do not
want to hear from you for the rest of the day. Oh ... go add the
numbers from one to one hundred.” So Carl grabbed his things and
plopped himself in the corner eager to meet the challenge.

“Now, if you really think about this, Carl could actually
be there all day. Carl did not have one of these,” I said,
holding up my graphing calculator .

“Is this a true story?” asked Megan.

I responded, “That is a secret. I could tell you but…”

“Then I’d have to kill yah,” the rest of the class yelled in
chorus.

“I want you to imagine you are Carl. You need to solve the
same problem that his teacher posed. There are a few rules however .
First, you may not use a calculator. Secondly, you can work
together if you wish. And lastly, you only have thirty seconds to
work on it.” I waited for the response.

“What?” “You’re kidding!” the students cried in disbelief.
I nodded. “After thirty seconds in the corner, Carl stood up
and said ‘I did it! I did it.’ Well, can you imagine the teacher’s reaction?
So, pretend you are Carl ; and yes, I am kidding, you have ten
minutes to find out how Carl did it in thirty seconds.”

Immediately the desks moved together, the talking commenced,
and they started to work, determined to solve the mysterious
problem.

You may have heard this story before; but students, most
likely, have not. If you have not, it is, for the most part, true. After
giving my students time to work on the problem, I reconvened the
class, and we discussed how Carl went about solving it. Carl is,
of course, Johann Carl Friedrich Gauss, the second most famous
mathematician ever to have lived.

After a discussion of the problem and their solutions, I continued
by showing the connection between it and the formula to sum a
series taught in high school mathematics curricula. The formula is:
where S_n is the sum of the series, n is the number of
terms in the series, a is the first term in the series, and t_n is the last
term in the series.

Carl discovered that the series 1 + 2 + 3 +. . . 98 + 99 + 100 has
a pattern. The first term + the last term = 101, the second term + the
second last term = 101, etc. Now, how many 101’s will there be?
50 pairs! Therefore, 50 x 101 = 5050. Applying this to our formula,
n = 100, a = 1 and tn = 100 giving

Before showing this formula to my students, I posed another
question al lowing them to apply “Carl’s logic” to a similar problem.
My students were then encouraged to apply this same principle to
every series! For example, given the series 9 + 11 + 13 + . . . + 41
that has 17 terms, the sum is Awesome!

Why the story? Is it a waste of valuable class time? I would
like to strongly argue that it is not a waste of time. In fact, I would
like to argue that taking the time to engage students in this manner
is vitally necessary!

I started by asking a series of questions. What does your mathematics
classroom look like? Does it look like the math classroom
you experienced through childhood? Are your students bored as you
were as a child? Are they engaged in the material presented? Do
they enjoy learning? Do they look forward to your class? Do they
participate in mathematical thinking and inquiry? Simply put, do
they like mathematics? The answers to these questions are of course
varied and complex . However, I hope you see that my students were
engaged in this lesson. They did enjoy learning! They did participate
in mathematical reasoning and thinking!

So what makes this class different from the norm? Can we use
this example to establish criteria that we can then use to engage
students and help them think mathematically? I believe we can.
Students must be encouraged to think for themselves by tapping
into their imaginations. Only when mathematical concepts are presented
in an imaginative way will students fully benefit from their
experience. Both teachers and students must engage themselves in
this creative realm of thinking, where imagination plays a vital part
in learning and teaching. As Egan (2005) states

Imagination is too often seen as something peripheral to
the core of education, something taken care of by allowing
students time to ‘express themselves’ in ‘the arts,’ while
the proper work of educating goes on in the sciences and
math… [But] imagination is the center of education; it
is…crucial to any subject, mathematics and science no less
than history and literature. Imagination can be the main
workhorse of effective learning [and teaching] if we yoke
it to education’s central tasks. (p. xiii)

By allowing my students to imaginatively enter Carl’s world
they became interested and involved. Students were intrigued.
They wanted to know more. They were not bored. “This concept of
teaching implies the need for teacher’s use of their own imagination
capacities while interacting with students to engage them in truly
enjoyable and relevant learning” (Jagla, 1994, p. 4). I was able to
turn a very theoretical math concept, summing a series of numbers,
into a meaningful, enjoyable, and relevant topic.

What were the characteristics of this particular class that
made it so effective in an imaginative way? Firstly, I allowed my
students to take part in the narrative. Students love to hear stories,
but they also like to reenact them. A “boring” concept was made to
come alive by engaging students in the story of “Carl,” and what
is especially fascinating to them is that the story was actually true!
Story telling can be used in any subject at any grade level, including
tenth grade mathematics. As Jagla (1994) observes, “Storytelling
is a delightfully imaginative activity for all ages…the telling of
stories is a wonderful way to provide context and make connections
for students at any level” (p.132). With a little effort, students were
able to put themselves in Carl’s classroom and picture themselves
in his predicament. The story included themes of hero and villain,
mystery, humor, and excitement. My students actually became part
of the narrative! And most of them solved the problem!

I believe students will remember this concept and how to use
this formula because it was given a personal dimension with the
introduction of Carl. The mathematical concept was reinforced
by the story. It would have been very easy to stand in front of
the class in a traditional manner and give them the formula with
three or four examples. Would this have been as effective? Would
students have been engaged, or would they have been bored, if
I had taught it in this way? Stories serve two purposes: they are
effective in communicating information and they orient the hearer’s
feelings about the information. The story of Carl communicated a
mathematical concept, but it also instilled positive feelings toward
the material that I was going to be testing them on the following week.

Secondly, this was an effective classroom experience because
students were able to discover the required learning outcomes
on their own. Through experimentation, most students were able
to solve the problem 1+2+3...+100. Some were able to apply it
to another example, and some even came up with a generalized
formula. Of course, this technique employs an old idea that has
been around for a long time: allowing students to discover material
on their own adds up to a rewarding educational experience.
Thinking skills are important to develop. With practice in problems
like the one that Carl solved, students are able to improve their
problem solving skills. It would have been very easy for me simply
to show them the concept; but students are much more likely to
own the material, understand it, remember it, and apply it to new
and different situations if they are given a chance to think. Herbert
Spencer puts it this way: “Children should be led to make their
own investigations, and to draw their own inferences. They should
be told as little as possible, and induced to discover as much
as possible” (Kazamias, 1966, p. 75). Encouraging the use of a
students’ imagination is directly related to critical thinking and
problem solving skills. Jagla agrees: “imagination is thinking of
the various possibilities of a certain situation…you have to know
scientifically what will work—
you also have to go after what has
never been done before, you don’t even know if it could work”
(1994, p. 30). My students had never seen a problem like this
before! By using their imagination, they were able to experiment
with different new ideas that led them to the solution.

Thirdly, Carl’s problem appealed to my students’ sense of
mystery and intrigue. My students, like Carl, were anxious and
excited to solve this problem on their own. They wanted to be
the “naughty boy” who was able to solve the problem the teacher
posed as punishment. How could Carl do this in thirty seconds?
Impossible! No, it was possible! Well then, prove it! My students
were engaged and became engrossed by the problem. It is an
awesome sight to see students become completely immersed
in a problem. And they solved the mystery! Math can actually
be interesting! As Egan (2005) states, “All the subjects of the
curriculum have mysteries attached to them, and part of our job in
making curriculum content known to students is to give them an
image of richer and deeper understanding that is there to draw their
minds into the adventure of learning” (p.6).

Lastly, another way to engage students in a topic is to use
humor. People love to laugh. When students enjoy themselves
they are more eager to participate, they are more likely to retain
and understand the material, and they are more likely to use the
concepts covered. Although it is not evident in my narrative,
humor was part of Carl’s story. In relating the story of Carl,
my interactions with my students involved humor. The general
presentation of the story can easily be made funny, thus increasing
the enjoyment of the experience. Humor can alleviate monotony
and boredom. Egan (2005) explains that “[It] can also assist in the
struggle against sclerosis of the imagination as students go through
their schooling—helping to fight against rigid conventional uses
of rules and showing students rich dimensions of knowledge and
encouraging flexibility of mind” (p. 4).

Are these the only techniques at our disposal to engage
students in learning? Of course not! If our chosen technique to
relay information to our students allows them to embrace the
material, then it is worthwhile. Narratives, self-discovery, a sense of
mystery, and the use of humor were four powerful tools used in this
particular situation. What other imaginative techniques could I have
employed in this lesson? Two other strategies that I often employ
involve role-playing and the use of mental imagery.

Let me offer another example. One of the most important
concepts taught in high school mathematics is factoring. The skill
of factoring is taught in every grade from nine to twelve. Less
complicated factoring is introduced in the elementary school grades.
To be successful in each of these courses, it is vital that students
understand how to factor, why to factor, when to factor, and what
to factor. For that reason, factoring is a huge “ production ” in my
classroom. If there is one thing my students remember about math
when they graduate it is the Wilke Bug.

The Wilke Bug is used as an ana logy for factoring . In grade
nine, the fundamentals of factoring are learned. Students begin by
learning common factoring . Given any polynomial , the first step
is always to look for a common factor; for example, 3x²+6x can
be factored into the form 3x(x+2) because the factor 3x is shared
between the two terms. After common factoring, there are two
other main types of factoring: difference of squares and trinomials.
Difference of squares has two terms, such as y²-16. This can be
factored into (y + 4)(y – 4) since its expansion is its corresponding
multiplication question . Trinomial factoring has three terms—for
example, x²+5x+6. This can be factored into (x + 2)(x + 3) for the
same reason. (Just as +2 and +3 multiplied is +6; and added is +5).
So, where and when does the Wilke Bug roll into town?

The procedure for factoring is compared to the process of
driving from town to town using a map to guide the steps. The
road first leads to the town of Common Factorville. It is always
the first stop! If a common factor exists between the terms of the
polynomial, it must be removed! I explain to my students that the
journey does not end at Factorville. “Keep driving” is a phrase in
common use in my classroom. However, after Factorville, the road
forks and students must then choose which path to take: two terms
leads to the town of Difference of Squaresville, while three terms
leads to Trinomialville. Understanding this concept is important
because some factoring questions involve both types; for example,
2x²–2 is first factored using common factoring 2(x²–1), before one
continues to Difference of Squaresville” 2(x + 1)(x – 1).

So why bother with such an elaborate metaphor to
explain a mathematical operation ? Wouldn’t it have been more
straightforward just to show the students the three methods and
then assign them practice questions? Maybe! But, for some reason
students love the whole Wilke Bug experience! Past students walk
by me in the hallway and ask me “How’s the Wilke Bug?” Some
students want me to get a license plate with the letters WILKBG
on my car! Each year, I have a Wilke Bug contest in my math
classes. The person that designs the best Wilke Bug receives bonus
marks! (See the attached picture for last year’s winning entry). In
higher grades I introduce students to other types of factoring, and
I add more paths to the fork in the road. I believe that this type of
activity contributes to an enjoyable, exciting, and unique learning
atmosphere. Students appreciate that there’s a serious side to

Students use a factoring map to drive the Wilke Bug through the process of factoring. I draw the factoring map as follows:

having fun in the class—they are actually learning one of the most
important concepts in high school mathematics.

The use of metaphors in the classroom is an effective tool that
teachers have at their disposal in any grade or subject. Egan (2005)
lists metaphor as one of the cognitive tools used to benefit student
learning. The Wilke Bug is an example that plays upon a student’s
(and teacher’s) imagination. “Metaphor is the capacity, or cognitive
tool, that enables people to see one thing in terms of another”
(Egan, 2005, p. 13). In this example, students use a simple map to
guide them through the journey of factoring. Egan (2005) argues
that, “The use of appropriate metaphors can stimulate the imagination
and creativity in all subject areas” (p. 13).

Another cognitive tool that I frequently employ when I teach
factoring is the use of role-playing. After the whole factoring
routine is explained, my class practices questions using the Wilke
Bug. I set up two chairs in the front of the classroom and ask for a
volunteer. Together, we drive the Wilke Bug around the classroom
using our map to guide us to the different towns. I start off by
calling shotgun. The use of this type of role-playing is an extremely
effective teaching technique, and it can be employed in any subject
and grade level. Role-playing engages students in a physical manner
and helps students to learn and retain information. Students
are captivated when material is presented in a different, often
outrageous manner. As Egan (2005) states, “there are endless ways
to shift the context so that the routine classroom becomes a place
where students never quite know what to expect… the imagination
can transform the classroom” (p.105). In my classroom I sometimes
feel that I am more of an actor than a teacher. Acting out the Wilke
Bug routine is part of the math show!

Another important tool for learning used throughout these
classes is mental imagery. I ask students to picture in their minds
the crazy Mr. Wilke driving the Wilke Bug around the classroom
when they are at work on factoring questions. This helps them to
remember the routine. These images provoke positive and humorous
feelings, while contributing to their understanding of the subject
material. In mathematics, it is extremely important to visually
picture ideas and strategies to various problems. Talking through
questions can sometimes help struggling students.

When it comes to evaluating students on factoring, I ask them
to write a factoring letter. They are to write a letter to anyone they
wish explaining the factoring process, so that the reader will be able
to do factoring. The letter allows the student to go through factoring
in a manner that works for them. Writing down the routine allows
them to sort through difficulties and put the procedure in their own
words. In other words it helps reinforce the idea.

Carl’s series and the factoring Wilke Bug are two examples
of the use of imagination in teaching mathematics. As educators it
is our responsibility to engage students in meaningful and exciting
lessons that aid in learning course material. The lessons are imaginative
in that they do just that. There are many techniques to present
material imaginatively, and it is the teacher’s task to develop
various activities that challenge students to think in creative and
innovative ways. The lesson about Carl employs storytelling, discovery
learning, mystery, and humor to achieve learning outcomes
effectively. The Wilke Bug makes use of role-playing, metaphors,
and imagery. These are a few of the imaginative learning tools at
our disposal. Be creative and have fun!