Since you are reading this short article, I assume that you are involved with and/or interested in math education. Can you name some of the people who have made important contributions to this area?

This* Information Age Education Blog* entry names two of my favorites, George Polya and Seymour Papert.

**George Polya**

**“A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.” (George Polya; Hungarian mathematician; 1887-1985).**

Polya’s focus in math education was on problem solving. Here is a quote from a talk he gave to a group of elementary school teachers (Polya, circa 1969):

To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics?** In the first place it means to be able to solve mathematical problems.** For the higher aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. [Bold added for emphasis.]

**Seymour Papert**

**“Nothing could be more absurd than an experiment in which computers are placed in a classroom where nothing else is changed.” (Seymour Papert; South African/American mathematician, computer scientist, and educator; 1928-2016.)**

Logo is an educational programming language designed in 1967 by Daniel G. Bobrow, Wally Feurzeig, **Seymour Papert**, and Cynthia Solomon. Papert is best known for his role in helping to design the Logo programming language, and then widely publicizing it through his talks and writing.

Seymour Papert’s education included two doctorates in mathematics, and working with Jean Piaget for a number of years. The work with Piaget developed Papert’s interests in how children learn and in creating environments that help children to learn. Computers and the Logo programming language provide an excellent environment in which children can learn problem solving.

Alan Kay is credited with inventing the laptop computer and has made many contributions to the field of computers in education (Kay, 2016). If you start at minute 16 of the video, *Alan Kay,* you will hear a short presentation by Alan Kay about a major contribution Papert made to his life (MIT Media Lab, 8/1/2016).

The Logo programming language puts a very powerful tool for creation and problem solving into the hands of very young students. Papert explains this idea in the first half-hour of one of his video presentations (Papert & Jettinghoff, 4/28/2004). Also see 10 instructional videos, Seymour Papert: On Logo (Papert, 1986.)

The fundamental idea underlying Logo is to provide an environment in which students can attempt to solve challenging problems, may make mistakes (think in terms of trial and error), and are challenged to find and correct their mistakes. Programmers use the word *debugging* to mean finding and correcting errors.

Debugging is an important concept applicable to all areas of problem solving. For example, as I write this document using a word processor, I make keyboarding and spelling errors. The computer can help me locate these errors, and then I can easily correct them. But, of greater importance is that I can look back over what I have written and think carefully about whether it effectively communicates my message. I can engage in the activity *revise, revise, revise* that is essential to good writing.

**What You Can Do**

While Seymour Papert and George Polya are very important math educators, the ideas of problem solving and debugging they focused on in their professional careers are of much greater importance. Problem solving lies at the very heart of every academic discipline. Over the centuries, humans have developed aids to representing and solving problems. Mathematics and computers are two examples of very powerful aids to representing and solving a very wide range of problems. Thus, when you help yourself and others to learn math, make sure you repeatedly emphasize these two ideas!

**References and Resources**

Kay, A. (2016). Alan Kay.* IAE-pedia.* Retrieved 3/4/2017 from http://iae-pedia.org/Alan_Kay.

MIT Media Lab (8/1/2016). Learning from Seymour Papert. (Video, 1h 2m.) Retrieved 3/4/2017 from https://www.youtube.com/watch?v=Pvgef9ABDUc. This site includes links to several presentations by Seymour Papert.

Moursund, D. (2016). Seymour Papert. *IAE-pedia.* Retrieved 3/4/2017 from http://iae-pedia.org/Seymour_Papert.

Papert, S., & Jettinghoff, R. (4/28/2004).Talk by Seymour Papert & Robin Jettinghoff. (Video, 1h 16m.) Retrieved 3/4/2017 from https://www.youtube.com/watch?v=4iIqLc0sjjs.

Papert, S. (1986). Seymour Parert: On Logo. (10 videos). Retroeved 3/8/2017 from http://el.media.mit.edu/logo-foundation/resources/onlogo/index.html.

Polya, G. (circa 1969). The goals of math education. Retrieved 3/4/2017 from http://mathforum.org/kb/servlet/JiveServlet/download/204-474401-1454303-72125/att1.html.

University of California, Berkeley (n.d.). Polya’s problem solving techniques. Retrieved 3/4/2017 from https://math.berkeley.edu/~gmelvin/polya.pdf.

Wikipedia (2017). Logo (programming language). Retrieved 3/4/2017 from https://en.wikipedia.org/wiki/Logo_(programming_language).

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