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"Puzzling through problems” as a habit of mind

Presenter(s): 
Paul Goldenberg
Meeting: 
32nd Panama conference
Date: 
January 16, 2014
Location: 
Noordwijkerhout, Netherlands

We often think of puzzles as special-purpose artifacts crafted solely for the purpose of entertainment. In many ways, though, mathematicians treat the problems they are attempting to solve—problems that require highly specialized background and sophisticated thinking and technique—in much the way that non-mathematicians treat puzzles. Importantly, it is quite different from the way that many students perceive “mathematics problems” in school. It is only a slight exaggeration to say that, when students encounter a problem that they do not know how to solve, they feel like it is someone’s “fault”: often, they feel that they, themselves, are just not good enough at mathematics; but they might also feel like the teacher didn’t adequately prepare them. With puzzles—at least with the ones that we take on voluntarily—we all feel quite differently. If, at first, we have no clue how to start, we look for that clue. We’re not the least bit surprised: after all, a puzzle is supposed to be puzzling! We don’t expect it to be a routine exercise that we already know exactly how to start and finish. Even a crossword puzzle with no mathematical content at all is a good example of such a “mathematical” approach. A challenging crossword puzzle for adults might have 60 to 80 clues. When you start the puzzle, most of these clues are useless! You need to look around, find something you can do, check it a bit, and then look around some more. Each step is not just “part of the job done”; it provides new information, changing the meaning and usefulness of other clues. Puzzles that do have mathematical content, unlike crossword puzzles, can be a powerful vehicle not only for that content, but for the way of thinking that serious researchers in mathematics and other fields use. The parameters of their difficulty—cognitive demand and mathematical prerequisites—can also be manipulated independently, so true challenges can be given to students with weak skills, and “easy puzzles” can be given to students with advanced skills.

This presentation will give examples and some of the rationale behind activities that teach this “puzzling through” habit of mind, and can challenge the best students without excluding others. Not all of them are “puzzles” in the usual sense—for one example, we will look at word problems without the question part!—but all of them involve some puzzle-like element that is key to mathematical thinking.
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  • And for readers puzzled by the name "Panama," it is an acronym: Pedagogische Academy Nascholing Mathematische Activiteiten.
Citations: 

Goldenberg, P., Mark, J., and A. Cuoco. An algebraic-habits-of-mind perspective on elementary school. In Hirsch, B Reys, and Lappan, eds, Curriculum Issues in an Era of Common Core State Standards for Mathematics, Reston: NCTM. 2012. (First appeared in Teaching Children Mathematics 16(9) pp. 548-556, May 2010. Reston, VA: NCTM. 2010.) Cuoco, A., Goldenberg, P., and June Mark. Organizing a curriculum around mathematical habits of mind. In Hirsch, B Reys, and Lappan, eds, Curriculum Issues in an Era of Common Core State Standards for Mathematics, Reston: NCTM. 2012. (First appeared in Mathematics Teacher 103(9) pp. 682-688, May 2010. Reston, VA: NCTM. 2010.) Mark, J., Cuoco, A., Goldenberg, P., and S. Sword. Developing mathematical habits of mind. In Hirsch, B Reys, and Lappan, eds, Curriculum Issues in an Era of Common Core State Standards for Mathematics, Reston: NCTM. 2012. (First appeared in Mathematics Teaching in the Middle School 15(9) pp. 505-509, May 2010. Reston, VA: NCTM. 2010.) Goldenberg, P., & N. Shteingold. “Early Algebra: The MW Perspective.” In Blanton and Kaput, eds. Algebra in the Early Grades. Hillsdale, NJ: Erlbaum. 2007. Goldenberg, P., & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: PreKindergarten–Grade 6. Reston, VA: NCTM. 2003. Cuoco, A., Goldenberg, P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996.