Teaching is an inherently wicked problem. (This idea has been addressed multiple times in this blog– search for “wicked.”) In 1973, scholars Rittel and Webber defined wicked problems as those that are ill-defined and that are judged only from the perspective of the individual who experienced the solution. A defining characteristic of wicked problems is that there is “no right to be wrong” as the solution will have permanent effects on those who experience it. Rittel and Webber observed, “every implemented solution is consequential. It leaves ‘traces’ that cannot be undone” (p. 163). Consider how this idea affects our understanding of teaching mathematics. When a student experiences a teachers’ lesson it will increase or decrease or leave unchanged the students’ ability to do the taught math. Further is could leave increase or decrease or leave unchanged the student’s ability to use the math and to learn more math.
The experience of a lesson is multidimensional as well as it will affect each student’s cognitive abilities to do math, their mathematical habits of mind, and their attitude towards math and their own abilities to do math. Further complicating the teaching and learning of mathematics is the fact that the lessons learned by and individual may be the one intended, the intended one incorrectly, or the incorrect ones.
In the 1970’s “Benny” demonstrated the disparities between students mathematical behaviors and their understanding of mathematics.
Reference
Rittel, H. W., & Webber, M. M. (1973). Dilemmas in a general theory of planning. Policy sciences, 4(2), 155-169.