Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

The First Proofs

July 2001

While preparing for the June's column I came across a collection Is "research in mathematics education" possible? by H. Wilf of University of Pennsylvania. (I've been searching the Web for information on the Parrondo paradox, about which Doron Zeilberger from Temple University has co-authored with his computer a short note and a Maple package. Wilf and Zeilberger shared the 1998 Leroy P. Steele Prize. So one thing led to another and eventually to the aforementioned page.) The collection deals mostly with accidental and deliberate misuse of statistics in math education research down to the faulty setup of experiments.

Around that time I've been reading Children's Mathematics - Cognitively Guided Instruction by T. P. Carpenter et al (NCTM, 1999.) Cognitively Guided Instruction (CGI) builds on the recognition that children tend to develop informal, i.e. with no formal training, counting strategies that differ from child to child. Children form their strategies in accordance with their experience and abilities. CGI aims to expand children's knowledge and practices on the individual level with what comes to them naturally. The book is a summary of a multi year research and practices of CGI.

The book is clearly weak on statistical side. Amidst enthusiastic teacher pronouncements and confidence building statements, like the very last paragraph

Thus, our studies consistently demonstrate that CGI students show significant gains in problem solving. These gains reflect the emphasis on problem solving in CGI classes. On the other hand, in spite of the decreased emphasis on drill and practice, there is no commensurate loss in skills.

the book offers no statistical data whatsoever, while I do not have an easy access to the references to verify the claims. The underlying idea, however, that children may be helped to develop their own problem solving approaches instead of being forced into some standard methods, is very powerful. The book's annunciation and subsequent classification of various children's strategies may provide a useful resource for teachers interested in employing the CGI approach in their classrooms.

As a sample of math education research, the book, statistical shortcomings aside, may face serious criticism on several other grounds. My main contention is that the reported research might have been boring to children. It might have planted the seeds of misunderstanding of what mathematics is about. Here's a sample of problems from the book:

 p. 3 Eliz has 3 dollars to buy cookies. How many more dollars does she need to earn to have 8 dollars? p. 21 Robin had 8 toy cars. Her parents gave her some more toy cars for her birthday. Then she had 13 toy cars. How many toy cars did her parents give her? ... ... p. 69 The elephant had 407 peanuts. She ate 129 of them. How many peanuts did the elephant have left?

Problem variations like these could help students grasp the gist of the problem, the underlying mathematical relation that leads to a successful solution. But no, in the book there is no pedagogical motivation. The authors simply used on different occasions differently worded problems.

Drill these problems may be not, but practice? Of course CGI students practice a lot. Moreover, they mostly practice solving the same kind of problems with modifications in the level of difficulty conditioned by the varying magnitude of numbers involved. Is it really what we mean by developing problem solving skills? I think not.

Problem solving skills are manifest in the degree of confidence with which students face novel problems. However, problemwise, the book (and apparently the research) only covered the traditional word problems from the standard number sense curriculum: four arithmetic operations and base-ten calculations. The variety comes from progressing from one digit number facts to double and triple digit operations.

As the book reports, children develop their own problem solving strategies. For example (p. 3),

... Zena, another first-grade student, could not recall the number fact 6 + 8 in solving a problem, but she knew that 7 + 7 is 14, so she said "I take 1 from the 8 and give it to the 6. That makes 7 and 7, and that's 14."

I believe that children who, like Zena, realize that counting is based on the first principle that associates a unique number with every finite set, are ready to try their hand, that is their mind, with less routine problems. These children are ready to argue, not just calculate. They are ready to prove. What can we offer them?

There are many problems that would fit young children's development.

For example, Scoring with a simple chip offers an opportunity to practice arithmetic skills and develop a sense for the modulo arithmetic. Subtraction games also offer good practice in number manipulations. The best part is that the two are equivalent. I think that being able to establish problem isomorphism is one of the most valuable skills in the problem solving variety. The Silver Dollar Game With No Silver Dollar is equivalent to Northcott's Nim. Puzzles like Four Knights, Looping Chips, or Seven Coins, are easily solved once their "other" representation comes to light. There is a good deal of problems built on the idea of invariance.

Here is a problem that might be called Sperner's Lemma for polygons.

Form a triangle with three small pins. Each pin has two ends. One is blunt, the other is pointed. The blunt end is known as the head, the pointed one is the tail. At a vertex of the triangle where two pins meet, they may meet either head to head, or head to tail, or tail to tail. There are just three possible vertex configuration.

The applet below helps one experiment with various configurations and even notice and prove by enumeration of all possible cases a nice fact that either all vertex configurations are the same or all are different. (With a click on a pin you can change that pin's orientation.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Now, what about four pins? With four pins, it's possible to have all vertex configurations equal. But since there are 4 vertices and only 3 possible configurations, it's impossible to have all 4 configurations different. Does that mean that we can't make a meaningful statement for 4 or more pins that generalizes that nice fact? No, it does not. There exists a very nice generalization, and the applet above may help formulate it as well. Just note that when you click on (or drag the cursor over) the number, i.e. the number of pins in the lower left corner of the applet, the number changes. To increase the number keep the cursor a little to the right of its vertical central line, to decrease it keep the cursor to the left.

There are many more problems that could be profitably tackled by young children. The lucky (in my view) ones will learn mathematics while playing games and solving puzzles. Others will be subject to other kinds of education. The field (I would not yet call it science) of math education is really a tough battlefield. Unlike mathematics, everyone seems to know something about it - even children. If you think that opinions come with age and experience, think again. Here's a message I received not long ago: