Setting the Right Tone in Mathematics for Future Elementary School Teachers
by Blair F. Madore AND Cheryl Chute Miller, State University of New York
The use of patterns and communication to solve mathematical problems is
a major theme in most elementary education programs. For the first
mathematics course in such a program another critical theme is how to replace
the fear of mathematics, in these future teachers, with enjoyment and self-confidence.
Many prospective elementary school teachers suffer from a variety of mathematical
anxieties and thus they need to know right away that this class will be
different from their previous “bad” experiences in mathematics. The activity
presented here has been used as a first day activity in the first mathematics
course for elementary school teachers at SUNY Potsdam. It was created by
Blair Madore with help from Victoria Klawitter, and field tested in various
stages of improvement by Laura Person, Uma Iyer, and Cheryl Miller. Working
together we (Miller and Madore) refined it into the activity presented
here. Its goal is to “set the right tone,” showing that the course
will be both fun and demanding, and that what seems impossible at first
can be solved when working together.
When used at SUNY Potsdam students sit in groups of four around square
tables. Each table is given a set of four cards (see the figure below).
Note that the cards are in a specific order and are colored (which we have
indicated here with the color word).
The only instructions given to the students are:
We divulge the solution as we discuss classroom reactions.
Draw the next figure of the pattern on each card, and answer the question
on each card.
The answers to the questions form a pattern; what should the next one be?
The cards themselves form a pattern. Determine what the next complete CARD
Student Reaction and Faculty Guidance
As we provide no rules for how the activity is to proceed, the first problem
for the students is to manage their behavior as a group. Most groups distribute
the cards, one per student, and each student solves one of the individual
problems. At many tables the students check each other’s solutions and
make corrections. They all begin to work together (if they haven’t already)
when confronted with instruction #2. Some struggle to understand it, though
most quickly see the sequence (10, 18, 22, 24). As the students at a table
work to find the next term, many must go back and check their answers from
each card to see if there was a mistake. Eventually everyone finds the
next term in the sequence and (incorrectly) presumes they are finished.
“That’s all there is to it, right?” a hopeful student at each
table will ask. They need to be reminded of instruction #3.
Maybe one third of the students understand what is being asked at this
point, while others look for a way the cards fit together to form a larger
figure. They often require help at this point such as series of oral
fill in the blanks: The next term in a sequence of numbers is a _________
(number – they reply in unison). The next term in a sequence of polygons
is a ________ (polygon). The next term in a sequence of cards is a __________
This is the low point of the class. The students felt great, expecting
that this class was going to be easy after all, because they had solved
the problems and even found the next term in the sequence. But now a daunting
task faces them, trying to understand what the question is really asking
them to do! We let them struggle and convince them not to give up, and
eventually they begin to discover the key questions:
Here the progress becomes much less uniform. Often for 10 to 15 minutes
no group makes progress independently, or a single group is able to make
headway. Usually at least one feature of the card is discovered, typically
either the shape or color, and the idea spreads through the class like
What features of these cards are significant?
What color should the next card be?
What shape should the next card be?
What should the card have on it? WHAT?!! You mean I have to make up a pattern
What should the question ask?
To help students, at this point, various hints can be given, such as:
Often the process of making a table describing the features of the original
four cards can help students discover the patterns.
Do the colors form a pattern?
How many sides does the first card have? What color is it?
Is there some relationship?
What kind of question should be on the card?
What shape should the dot patterns have on the new card?
There are at least 6 different patterns going on here!
Table 1: Summary
In most of the classes at least one group discovers a complete and correct
Answers vs. Good Answers
One important idea that this puzzle naturally brings out is the difference
between a solution and a “good” solution, i.e. one that is clear and leaves
little doubt. The easiest way to see the difference is in how the
shape of the card is determined. Looking at the table above many
students decide that the shape is a square in many ways. Some conclude
it is a square since that is the “only shape not mentioned”. This
can prompt the instructor to talk about other shapes such as a trapezoid,
a rectangle, an octagon, or others. Eventually the student realizes
that even though “square” is correct, the explanation behind it does not
make this a “good” solution since there are doubts about the reasoning.
In a similar way if they look at the number of sides (3, 6, 6, 5) students
often say “it must have 4 sides since that number has not been used yet.”
Again the question of why must 4 must be used leads to a similar discussion.
“Is there a reason we must use all of the natural numbers? Why can’t
4 be skipped? Why shouldn’t 5 occur twice in a row like 6 does?”
Again, the explanation leaves doubts. When the students eventually discover
the relationship between the color of the card and how many sides it has,
the answer becomes clearer. The blue card (the colors follow the
colors of the rainbow ROYGBIV) must have 4 sides as blue has 4 letters.
As all of the shapes are regular polygons a square is the polygon we should
use. Thus they have found a “good” answer.
Sometimes a student (or a faculty member) will point out that the entire
problem is based on the order of the colors of the rainbow – something
that is not universally agreed upon from country to country or language
to language. Thus, is this really a “good” answer? This is a valid point
but one that does not generally bother our students. It does provide an
opening for a discussion of how to avoid alienating segments of your class
and being culturally sensitive.
The use of an ice-breaker on the first day of class helps students get
to know each other as well as the instructor. With this puzzle the
fear of being in a math class can be lessened while students learn that
they can solve difficult problems by talking together. Most of the students
never knew that so much of mathematics deals with patterns, the main idea
of this course and the first topic in their textbook . As one student
commented “in this course the most interesting thing was finding patterns...and
now I see or try to look for patterns in so many things.”
Setting the correct tone, and maintaining it throughout the semester,
encourages significant mathematical development in students who were initially
unsure of their own abilities. The students also developed the confidence
and positive attitude needed to tackle more complicated mathematical ideas.
Students learn quickly that although the course will make them think, and
think hard, about nontrivial mathematics, this time they can succeed and
1. Billstein, R., Libeskind, S. and Lott, J.W, 2004, A Problem
Solving Approach to Mathematics for Elementary School Teachers, 8th
edition (Boston: Pearson Education).
Blair Madore (email@example.com)
is a native of Newfoundland, Canada and holds a BMATH from the University
of Waterloo and M.Sc and Ph.D. from the University of Toronto. His research
area is ergodic theory. He is currently an Associate Professor of Mathematics
at the State University of New York at Potsdam.
He has been very active in educating future teachers and is passionate
about mathematical outreach.
Cheryl Chute Miller (firstname.lastname@example.org)
received her B.S. in Mathematics from John Carroll University, near Cleveland
Ohio, and Ph.D. at Wesleyan University in Middletown Connecticut.
She has been teaching at the State University of New York at Potsdam
ever since and is currently a Professor of Mathematics. Her research area
is model theory, but she has strong interests in the history of mathematics.
As a faculty member at SUNY Potsdam she has helped in the hard work of
developing teachers at all levels.
Both can be contacted by mail at: Department of Mathematics, State
University of New York at Potsdam, Potsdam, New York 13676-2294.
Teaching Exchange is edited by Bonnie