Some thoughts on Problem Posing Sandboxes [a work in process] Before anything else in mathematics there is size [magnitude]. Before anything else in language there are nouns, both count nouns and mass nouns. [Count nouns provide an answer to the question "how many?" - mass nouns to the question "how much?"] Think of a mathematical Problem Posing Sandbox as an arena for exploration and inquiry - allowing, indeed inviting, us to ask interesting "how many?" and "how much?" questions about the world around us. Seen in that light some possible sandboxes include a ball point pen, a kitchen sponge, your hair, and the people of Boston. Here, for example and by way of illustration, are some unusual questions about length - how long a line can a ball point pen write? area - what is the surface area of a kitchen sponge? volume - what is the volume of that thundercloud? weight - how much does a full passenger jet airplane weigh? time - how long does it take you to eat your weight in food? speed - how fast does your hair grow in miles/hour? number - how many people are there in Boston at this very minute? These are all quantitative questions. These questions all have quantitative answers. None of these questions has a precise quantitative answer. Yet, clearly each question has a "reasonably correct" answer. Here is another example - this time all the questions are prompted by a single setting - i.e., one million dollars. I think of the one million dollars as an arena for the posing of interesting "how many?" and "how much?" questions. I think of it as a problem posing sandbox. length - how far would $1M in one dollar bills laid end-to-end reach? in one hundred dollar bills? area - what is the largest rectangular area $1M in one dollar bills can cover
if laid out on the ground next to one another without overlapping? in one hundred dollar bills? weight - what does $1M in one dollar bills weigh? could you carry it? in one hundred dollars bills? could you carry it? time - at the rate of $10/hour how long does it take to earn $1M? how many {weeks, months, years} is that? number - how many one dollar bills can you fit in your backpack? You may regard some of these questions as playful and impractical - indeed some are - but there is no denying that people need to know about the magnitudes of attributes of length, weight, time, speed etc. in the world around us. Grown people are generally between five and one half and six and one half feet tall - not eight feet tall, a gallon of milk weighs about eight pounds - not eight ounces, the refrigerator in the kitchen holds about twenty cubic feet - not fifty, fifty miles per hour is a reasonable highway speed - not 5 m/h and not 100 m/h, etc. Such quantities are the stuff of human activity. Questions such as the ones above stand in sharp contrast to the sorts of questions traditionally posed in mathematics classrooms around world. School math questions demand unique correct answers - and the questions we generally pose in our classes in fact do have unique correct answers. This is odd because ostensibly a major reason for the presence of mathematics in the curriculum of our schools is to equip youngsters with quantitative and other analytic tools for dealing with their world. For the most part, the world people live in demands "reasonably correct" answers to quantitative problems. At the very heart of dealing with the world quantitatively is modeling - an activity whose essential elements include identifying and acquiring needed data, and combining these data appropriately to shed light on the situation being analyzed [1]. These are the very elements of the sorts of tasks I am proposing that we make much wider use of in the mathematics curriculum. It is clear that as a society we have, by and large, not succeeded in persuading the majority of our students of the importance of mathematics in their lives. Could it be that students understand intuitively that, for the most part, size and magnitude in the real world can only be determined approximately and that important sizes and magnitudes are rarely, if ever, known precisely? I do not wish to be understood as saying that learning the computational and other manipulative skills needed to compute correct answers is not necessary. I believe these skills are indeed necessary but need to be considered in the context of the larger world in which they will be put to use. The world around us needs answers that are "reasonably correct". That means that we must develop an appreciation of the context in which we apply our school learned skills as well as some taste and judgment about when, how and to what extent we use those skills. Is there anything special about the attributes listed above - i.e., length, area, volume, weight, time, speed and number - that mathematics educators should take into account in formulating questions and posing problems that have "reasonably correct" answers? I would like to suggest, at least tentatively, that the answer is yes. Humans have innate perceptual apparatus to detect presence and/or absence of these attributes and the ability, by virtue of their anatomy and physiology, to roughly order their magnitudes. It is possible to think that quantitative questions that speak to the human sensory apparatus might help us help our students come to understand that mathematics can be useful, and possibly even interesting. [1] See, for example, Schwartz, J.L., "Formulating Measures: Toward Modeling in the K-12 STEM Curriculum" in Brizuela, B.M. & Gravel, B.E., eds. "Show me what you know: Exploring Student Representations Across STEM Disciplines", 2013, Teachers' College Press, New York Here is a potpourri of problem suggesting/provoking/posing settings that may serve as sandboxes for the imaginative teacher - Hoops ! - a sandbox for basketball related questions The Car in Question - two links [ car 1, car 2 ] to applets that serve as sandboxes for the posing of interesting automobile related questions. A "Matched pair" of problem posing sandboxes - two applets that are based on settings that have similar mathematical structure - A travel sandbox for early algebra motion problems - Some WIHAW [ What Is Happening And Why ] sandboxes - Sandboxes for generating two standard types of word problems Age Problem Calculator - an applet that is designed to allow you to formulate word problems about age Coin Problem Workbench - an applet that is designed to allow you to formulate word problems about age ...and finally In February 2015 I was asked to design a collection of applets for a new version of "E15 - Introduction to the Calculus A", a course at the Harvard Extension School. The instructor for whom the applets were being designed wanted to pose his own questions to accompany the applets. As a result the collection of applets I produced consists of interesting interactive situations to which I have not attached questions for exploration. I hope that these applets can serve to stimulate your imagination and creativity and be a springboard for you to pose questions of your own. Judah L. Schwartz  mathMINDhabits by Judah L. Schwartz is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |