I have always considered Judah L. Schwartz a true pioneer in the field of learning technologies. He had the ability to look at the world through the new and different lens of the computer. He looked at the ethical and philosophical issues arising from the use of technology in education. His research interests include the use of the digital world to improve the teaching and learning of math and science. He designed the Geometric Supposer series of software and What Do You Do with a Broken Calculator? and other alternative software programs.
He is currently Professor of the Practice and Research Professor of Physics and Astronomy at Tufts University; he is also Emeritus Professor of Engineering Science and Education at MIT and Emeritus Professor of Education at the Harvard Graduate School of Education.
He is a remarkable pioneer in our field because he saw technology as a way of looking at mathematics in very new and alternative ways. His software asks the learner to think over and over again, “What if?” What if I change this value. What happens? The computer allows for infinite changes and explorations of these alternative operations. He likes to say the Ptolemy observations of the solar system were accurate. There was just one thing wrong with them and that was they were basically incorrect.
I especially like the “broken calculator” because with a bit of thought the learner can work around the faulty keys on the calculator. I believe this gives the learner a set of strategies that opens creative thought patterns. It also introduces the concept that there is more than one way to solve a problem. Judah’s software offers alternative ways to solve mathematical problems and to allow the learner to suppose different alternatives.
Here’s an example. Divide the class into teams of four or five students. Each team has a calculator. As much as possible, you want every team member to participate. The calculator is broken, that is, only certain number keys work. All the calculation functions work, that is, they add, subtract, multiply and divide. Each team has an observer that will describe the team’s work. The teacher writes an answer on the board, and the teams begin work:
Number keys that work are only 1, 3 , 5, 8
The first team that comes up with the correct answer wins that round.
Answer: 8 X 8 X 8 = 512 + 513 = 1025 – 8 = 1017 - 3 = 1014 – 1 = 1013
Another quicker answer might be 1111 – 85 = 1026 – 13 = 1013
Teams can choose any strategy they want to arrive at their answers. If they win, their observer must explain their procedure.
When the Supposer software was released, I let a senior mathematician on another project borrow my copy. He said he stayed up all night exploring alternative problems. I think some of the genius behind the software is the fact that you can examine over and over again alternative problems.
Thank you, Judah, for your contributions.
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