Constructivism and More

Pi – Pe – Pa Principles

This past weekend I heard a powerful, inspirational presentation by a wise, older gentleman, Mr. Janzen, who has been a college president, a pastor, and a counselor. He talked about three principles that have guided him through the years. He calls them the pi – pe – pa principles, where pi stands for powerful insights, pe for painful effort, and pa for productive achievement. As he developed these principles with examples from his own life and the lives of a few others, I had to think about the work of the AIMS Center.

  • Powerful Insights
  • Painful Effort Productive
  • Achievement

In the Research DIvision of the AIMS Center, our primary mission is to translate research into classroom practice. This vision came out of a deep concern about student learning of fractions and the consequences of their knowledge too often being the result of rote learning. I remember how one of the experiences that helped to bring focus to the vision was our coming across a book by Les Steffe and John Olive titled, Children’s Fractional knowledge. The book represents the bringing together of 25 years of intense research by Professor Steffe, his colleagues, and graduate students. Looking back on that first encounter with Steffe’s research and his synthesis of that research, I realize that what we experienced was what Mr. Janzen would call Powerful Insights. What jumped out at us began in the first sentence of the first page of the Preface to the book. Here are the first two sentences:

“The basic hypothesis that guides our work is that children’s fractional knowledge can emerge as accommodations in their natural number knowing. This hypothesis is referred to as the reorganization hypothesis because if a new way of knowing is constructed using a previous way of knowing in a novel way, the new way of knowing can be regarded as a reorganization of the previous way of knowing.”

While the reorganization hypothesis caught our attention like nothing else we had found so far in looking at the literature related to the learning of fractions, we saw it as opening up a whole new way to approach children’s coming to have fractional knowledge.  We had no idea how much time and effort would be required to understand all that was meant by and implied by this “reorganization hypothesis,” let alone the effort that would be involved in an attempt to translate and implement this research.

For our senior and associate researchers, as we are working to deeply understand the Steffe research, its theory base, and the reorganization hypothesis, we find ourselves in the middle of that Painful Effort that Mr. Janzen referred to as his second principle. It turns out that before we could begin to think about the reorganizational hypothesis that facilitates the move from natural numbers and whole number sequences to fractions, we would have to understand years of research related to children’s knowledge of natural numbers and the whole number sequence. There were two books that pulled this earlier research together along with numerous articles. After two years we are just beginning to finally get to work on Children’s Fractional Knowledge. How about that for painful effort?

I believe we would all agree that the painful effort was also rewarding effort and that already we are beginning to sense some of that Productive Achievement that is Mr. Janzen’s third principle. This is the reward for having a powerful insight followed by painful effort. We have made great strides in understanding the research and we are beginning to translate what we are coming to know into classroom practice. What we really look forward to is one day seeing children who, beginning in pre-school, are building the foundation for acquiring the whole number sequence through the early grades and then seeing them experience the reorganization of their knowledge of the whole number sequence so that they use that knowledge to construct their fractional knowledge — their fractional concepts and operations. That is the ultimate Productive Achievement we are working toward.

Pi, Pe, Pa, three principles—powerful insights, painful effort, productive achievement. They seem like powerful principles to live by, whether in our professional lives or our personal lives.

A Dream Coming True!

This blog post is being written from Tucson, Arizona, where Tiffany Friesen, Paul Reimer, and I are attending the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. The approximately 600 men and women attending this conference are almost exclusively university professors along with their graduate students,… Continue Reading

A Juggling Act

In previous blog posts we have, in various ways, talked about the commitment of the AIMS Center to a constructivist understanding of how children come to know.  There are several reasons for this choice, but probably the most relevant is that the most significant and extensive research related to how children come to know whole… Continue Reading

Show Your Thinking

Recently I was speaking with a parent who was expressing frustration with their child’s work habits in their math class. It was a conversation I have experienced many times throughout my 20 years of teaching. It’s about that child who says they can do the math in their head and they do not need to… Continue Reading

Constructivism and More

The Research Division of the AIMS Center is organized into four teams, of which three teams are presently focused on research related to how children come to know number. Our ultimate goal is to translate that research into classroom practice.  The theory base underlying the research we are following is what might be called a… Continue Reading