
by Blair F. Madore and Cheryl Chute Miller, State University of New York at Potsdam

Introduction
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 selfconfidence. 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.
The Activity
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:

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 should be.
We divulge the solution as we discuss classroom reactions.
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 __________ (card).
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:

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 of dots?

What should the question ask?
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 wildfire.
To help students, at this point, various hints can be given, such as:

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!
Often the process of making a table describing the features of the original four cards can help students discover the patterns.
Table 1: Summary
Card Number

Color

Shape

Number answer

1

Red

Triangle

10

2

Orange

Hexagon

18

3

Yellow

Hexagon

22

4

Green

Pentagon

24

5

?

?

25

In most of the classes at least one group discovers a complete and correct final answer.
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.
Conclusions
The use of an icebreaker 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 [1]. 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 enjoy math.
Reference
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 (madorebf@potsdam.edu) 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 (millercc@potsdam.edu) 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 136762294.
The Innovative Teaching Exchange is edited by Bonnie Gold.