This interactive essay, still a work in progress, draws on several decades of experience working with teachers of mathematics. It is an attempt to distill what I have learned about teachers who make a deep impact on their students and teaching that can change the way both teachers and students think about the subject of mathematics and its applications.
In this essay, I will not address the larger issue of why mathematics is a part of the school curriculum. Nor will I address the question here of how to pose questions that will cause students to come to understand the importance of mathematics to their own lives. I have written elsewhere about this subject [1] and address it again in an essay on "Problem Posing Sandboxes" on the mathMIND habits website [url]https://sites.google.com/site/mathmindhabits/[/url].
My strategy for addressing this goal is to describe four themes that seem to me, both individually and collectively, important for the posing of problems in mathematics. I choose to illustrate these themes with small interactive environments [applets] and questions meant to instantiate the role of these themes in the posing of problems. The applets are all drawn from the mathMINDhabits website which was put together to support the course on problem posing that I have been teaching at Harvard for the past several years to middle and high school math teachers.
My starting point is a set of guidelines for good challenges that appears on the mathMINDhabits website
[b][i]A good challenge should
• be easy to get started on, drawing on likely knowledge and past experience of the student
• arouse curiosity – successful challenges often build on apparent inconsistencies or start from seemingly unintuitive premises
• offer an engaging environment in which the student is invited (and indeed, encouraged) to make conjectures
• invite extensions to more general and/or analogous situations
• not have a single correct answer nor should the student be tempted to approach it as if it did
Possible opening sentences for challenges include
• Why do you think…?
• Which of these explanations make more sense to you and why?
• If ______ is true, then how come _______ ?
• How do you think this situation does or does not resemble ________ ?
• What do you think might be true about this ________ ?
• What do you think this is a case of? [/i][/b]
All too often the problems and challenges we pose for our students to solve call for unique correct answers. I have always been uneasy about problems of this sort, feeling that they are formulaic in nature. In fact, for many years, I had hanging on the wall of my office at Harvard a sign that read
[i][b]“Avoid posing problems that have only ONE correct answer!” [/b][/i]
Adhering to this rule inevitably leads one in the direction of posing design and/or synthesis problems that by their very nature rarely have unique answers. Here is an example of how a problem posed in traditional form can be transformed into one having more than one answer.
Traditional form: Find the value of x that satisfies the equation 3x + 5 = 7x – 13.
Transformed form: Unsolve the equation x = 9/2 to show that 3x + 5 = 7x – 13. This unusual expression means build a sequence of equivalent equations linking the equation x = 9/2 and the equation 3x + 5 = 7x - 13.
The solution to the first problem is a single statement x = 9/2. The solution to the second problem is a sequence of equivalent equations beginning with x = 9/2 and ending with 3x + 5 = 7x – 13. Clearly such a sequence is not unique! A sequence of equations, if correctly generated, will consist of equations that all belong to the same equivalence class each of which can be shown to be equivalent to the starting and ending equation.
Similarly, rather than ask an algebra student to find the roots of a given quadratic, we might ask the student to find a quadratic whose roots lie between the roots of the given quadratic - rather than ask a geometry student to prove a given theorem about triangles, we might ask the student to generate conjectures about a given construction with triangles. There are countless (no pun intended) such opportunities in the elementary and secondary school curriculum.
In order to support the strategy of focusing on design/synthesis questions I feel that it is important to provide teachers and students with proper tools for the investigation of such questions. In my view, this means interactive software that is [i]exploratory[/i] in nature. Exploratory tools, in contrast to demonstrative tools, permit investigating phenomena rather than simply exhibiting them. This distinction will emerge clearly in the course of this essay.
There are four fundamental mathematical actions that serve as themes and shape the choice of the problems with “more than one correct answer” presented here. They are "Classifying, Ordering and Comparing, Representing, and Modeling and Formulating".
[b]Classifying[/b]
Any equation one writes can be said to belong to an “equivalence class” that consists of all the equations with the same solution set. Solving the equation can then be regard as a series of transformations that transform the original equation into another member of the same equivalence class that has the form x = {set of numbers} [i.e., a single number in the case of a linear equation].
The first illustrative example in this section “Solving Linear Equations Graphically & Symbolically” allows one to enter a linear equation in the form f(x) = g(x). Plots of the functions f(x) and g(x) appear on the screen and the solution x = a (say) is immediately apparent. Note also that the point {a, f(a)} is the same as the point {a, g(a)}.
The user is then offered the opportunity to generate other members of the equivalence class of this equation using symbolic transformations – [adding or subtracting a constant from both functions, adding or subtracting a linear term from both functions or multiplying both functions by a non-zero constant]. Each of these transformations produces another member of the same equivalence class of equations.
Alternatively the user may use graphical transformations – [sliding the point {a,f(a)} vertically along the line x = a, rotating f(x) about {a, f(a)}, or rotating g(x) about {a, g(a)}. Each of these transformations produces another member of the same equivalence class of equations. [i]The reader will note that the latter graphical transformations seem to be inconsistent with the way solving equations is normally taught where we insist that the same action must be carried out on [b]both[/b] sides of the equation. These last two transformations each transform only one side of the equation.[/i]
Do these transformations exhaust the set of all possible transformations that can transform one member of the equivalence class of equations into another? Do these transformations suffice to generate all the members of the equivalence class of equations?
[i][b]explore applet: [url]http://tube.geogebra.org/m/1719993[/url][/b][/i]
The remaining three illustrative examples in this section approach the study of polygons as a sequence of hinged links whose lengths and angles with neighboring links can be varied. The first of these “Quadrilateral factory?” poses the following questions “Break a stick into four pieces. Under what circumstances can you use the pieces to form a closed quadrilateral? and, “[i][b]If [/b][/i]you can form a quadrilateral, what classes of quadrilateral can you form?” [i]This problem is a generalization of the classic problem of breaking a stick into three pieces and asking about the likelihood of being able to form a triangle with those pieces. Because of the interactive nature of the environment, the problem is easily started and explored and the inquiry extended to other polygons.[/i]
[i][b]explore applet: [url]http://tube.geogebra.org/m/1858001[/url][/b][/i]
In “Polygon Explorer” polygons are represented by a histogram of ordered segment lengths divided by the perimeter of the polygons and by an ordered histogram of interior angles divided by the sum of the interior angles of the polygon. The two histograms serve to emphasize both nature of the class of polygons and its similarity to and differences with other classes. The spatial representation of the polygon and the histogram representation of the polygon are linked and both representations are controlled by dragging the vertices of the polygon in the spatial representation.
Different classes of polygons will be characterized by length and angle histograms that have properties that reflect the properties of the class of polygon [e.g. a kite histogram will have two adjacent pairs of equal size length bars and two alternating pairs of equal length angle bars.]
[i][b]explore applet: [url]http://tube.geogebra.org/m/2473747[/url][/b][/i]
In “Polygon Synthesizer” polygons are also represented by a histogram of ordered segment lengths divided by the perimeter of the polygons and by an ordered histogram of interior angles divided by the sum of the interior angles of the polygon. Here too, the two histograms serve to emphasize both nature of the class of polygons and its similarity to and differences with other classes. The spatial representation of the polygon and the histogram representation of the polygon are linked and both representations are controlled by varying the angles and link lengths in the histogram representation.
[i][b]explore applet: [url]http://tube.geogebra.org/m/2579167[/url][/b][/i]
[i]“Polygon Explorer” and “Polygon Synthesizer” are complementary examples – in one case the investigation is controlled by manipulating the spatial representation and the consequences of the manipulation are seen in the histogram representation – in the other case one controls the histogram representation and the consequences are seen in the spatial representation. The strategy of manipulating elements of a situation in one representation and asking students to anticipate the consequences of that manipulation in another representation in a central element of what I think of as [b]understanding[/b].[/i]
[b]Representing[/b]
In this section I continue the theme of using unfamiliar representations for familiar mathematical objects in order to get teachers and students to think differently and creatively about mathematics. Doing so often forces the teacher or the student to think across representations – internally posing the question what is the consequence in the other representation of a change of attribute in this representation.
The illustrative examples in this section are precisely of this nature.
Let me be quite specific – the linear function [b]y = mx + b[/b] can be represented as a line in the Cartesian {x,y} plane. It can also be represented as a point in an {m, b}, i.e. the {slope, intercept} plane. A linear function in the (x,y) plane is represented by a point in the (m,b) plane. It is also true that a linear function in the (m,b) plane determines a point in the (x,y) plane. Given a linear function in the (m,b) plane, what point does it define in the (x,y) plane?
[i][b]explore applet: [url]http://tube.geogebra.org/m/84147[/url][/b][/i]
Similarly, a time on an analog clock can be represented by a piecewise continuous function in an {angle, time} [position of the minute hand] plane along with a linear function in the {angle, time} plane for the hour hand. Is every point in this plane a legitimate time? Given two points that denote times, how can one determine the time interval between the times? [i]This example, as well as the following two examples, clearly draw on likely knowledge and familiar material cast in an unusual representation.[/i]
[i][b]explore applet: [url]http://tube.geogebra.org/m/2615311[/url][/b][/i]
A rational number, p/q, can be represented as a point in a {denominator, numerator} plane. In such a representation, traditional topics such as common denominators and common multiples display new and added richness.
[i][b]explore applet: [url]http://tube.geogebra.org/m/81486[/url][/b][/i]
Any isosceles triangle can be represented by a point in an {altitude, base} plane. The loci of equal area, equal perimeter and similar isosceles triangles are particularly interesting to study. [What functional form do the loci have?]
[i][b]explore applet: [url]http://tube.geogebra.org/m/93197[/url][/b][/i]
Any quadratic function of the form [b]y = x^2 + Px + Q[/b] can be represented by a point in the {P, Q} plane in addition to its graphical representation in the {x, y} plane. Some points in the {P, Q} plane correspond to quadratics with real roots and some to quadratics with complex conjugate roots. What regions of the {P, Q} plane do such points inhabit? What is the shape of the boundary(s) between the regions?
[i][b]explore applet: [url]http://tube.geogebra.org/m/156458[/url][/b][/i]
[b]Ordering & Comparing[/b]
Ordering a collection of related mathematical objects according to some criterion is a fundamental mathematical task. Here are four examples of problems in which correct answers span many values and thus present an opportunity for students to discuss their individual approaches to the problems. In the case of the first two illustrative examples, "Between-ness in Addition & Subtraction" and "Between-ness in Multiplication & Division" the mathematical objects in question are pairs of simple computation problems. The question posed in each instance - "Make up a problem of the same type whose answer lies between the answers to these two problems". Note that there are an infinite number of correct answers to any such problem as well as an infinite number of incorrect answers. Nonetheless, any given answer can be judged to be correct or incorrect. This diversity of possibilities allows for a good deal of individual expression and classroom discussion.
[i][b]explore these applets: [url]http://tube.geogebra.org/m/2703125[/url], [url]http://tube.geogebra.org/m/2703969[/url][/b][/i]
The next applet "Parabolas between Parabolas" carries the idea of "between-ness" into the realm of algebra. Users of the applet are asked to form two non-intersecting quadratic functions and then to explore the universe of parabolas that are everywhere larger than the smaller of the two parabolas and smaller than the larger of the two parabolas.
[i][b]explore applet: [url]http://tube.geogebra.org/m/2659337[/url][/b][/i]
The next applet in this section, "Area-perimeter workbench", tries to address the question of ordering when there is more than one attribute that can reasonably be ordered. This is the case when students are presented with collections of shapes in a plane and are asked to compare areas or compare perimeters. Students often confuse area and perimeter with one another. The applet provides an opportunity to build and explore two shapes with the property that the one with the larger area has the smaller perimeter and [i]vice versa[/i].
[i][b]explore applet: [url]http://tube.geogebra.org/m/2117365[/url][/b][/i]
The idea of “between-ness” as a guide to the posing of problems is one that can be used throughout the gamut of school mathematics. I have tried to illustrate its use in the arithmetic of the lower grades, in algebra and geometry. It is, however, more general than finding a one-dimensional range of values for which some property is true. It is a special case of the distinction between "inside" and "outside". Here, for example is a problem in which one is asked to find a two-dimensional region within which some property is true and outside of which it is not.
[b][i]In an equilateral triangle, the sum of the distances from any point in the interior to the sides of the triangle is constant. Clearly, under some circumstances the three distances can themselves form a new triangle (say the distances from the center of the equilateral triangle) and under other circumstances the three distances cannot form a new triangle (say the distances from a point very close to one of the vertices of the equilateral triangle). Find the shape and area of the region from which the distances to the sides of the equilateral triangle can form a new triangle.[/i][/b]
[i][b]explore this problem: [url]http://tube.geogebra.org/m/116064[/url][/b][/i]
A final remark about comparing - needless to say one must be careful in making comparisons not to try to compare things that cannot be compared - occasionally one sees questions of the form "Given a square of side [math]a[/math], for what value of [math]a[/math] is the area of the square equal to its perimeter?" - [depending on the choice of units, [i][b]a[/b][/i] can be any physical length], or "Given the function f(x)=b^x, for what value of b is the function equal to its derivative?" - [a question that makes no sense if x and f(x) refer to time, distance, density, population or anything else in the world around us].
[b]Modeling & Formulating[/b]
The first example in this section examines the issue of definition in mathematics. The remaining four of the five examples in this section focus on the most fundamental aspect of applying mathematics to the world around us, i.e. formulating a measure than can be incorporated into a model of how different aspects of the world are related. [2]
In the “Students define a median” illustrative example the following problem is presented –
Ninth grade students using a dynamic geometry program are asked to explore possible definitions of median in a triangle.
When the students choose a vertex and ask the program to produce a median, the program responds by drawing the median from that vertex.
Students explore the values of areas, angles and side lengths and suggest the following “definitions” –
[i][b]A – a median divides the side opposite the vertex in two equal pieces
B – a median divides the area of the triangle into two equal areas[/b][/i]
Can you show that these definitions are equivalent – if either one is true then the other is as well?
Why might definition A be preferable to definition B ? Why might definition B be preferable to definition A ?
Can the definition of median be generalized to polygons with more than three sides?
[i][b]explore applet: [url]http://tube.geogebra.org/m/2610055[/url][/b][/i]
The next applet “A Car & 20 Questions” explores the range of quantities that can be derived from reasonably well-known quantitative data about cars – average speed, average cost of a gallon of gasoline, average gasoline consumption and, perhaps less well-known, average weight of a gallon of gasoline.
[i][b]explore applet: [url]http://tube.geogebra.org/m/2002425[/url][/b][/i]
The next three applets are based on tasks that were developed by my colleagues and me in the Balanced Assessment in Mathematics Project at Harvard. They all deal with mathematizing a relationship that people are aware of perceptually but probably have never attempted to describe in any formal, not to mention quantitative, fashion. Each of these tasks, “Square-ness of Rectangles”, “Equilateral-ness of triangles”, and “Smooth-ness of Spheres” requires students to identify and describe formally a geometric property of some two or three dimensional shape. It is important to stress that properties such as “squareness” or “equilateral-ness” are not formal geometric properties. There are no formally correct and universally accepted answers to these questions. On the other hand, there are sensible (and non-sensible) answers.
[i][b]explore these applets:
[url]http://tube.geogebra.org/m/cIHqv4Ny[/url] ,
[url]http://tube.geogebra.org/m/84181[/url] ,
[url] http://tube.geogebra.org/m/209455[/url][/b][/i]
[b]Coda[/b]
Georg Cantor once wrote, “In mathematics, the art of proposing a question must be held of higher value than solving it.” In writing this essay I have attempted to explicate at least part of what may have been his intended meaning – i.e., design/synthesis questions often offer fruitful soil for those attempting to plant the seeds of exploration, inquiry and creativity in their students.
[1] [b][i]Can Technology Help Us Make the Mathematics Curriculum Intellectually Stimulating and Socially Responsible?, International Journal of Computers for Mathematics Learning, 1999, Vol. 4, Nos. 2-3, pp. 99-119[/i][/b]
[2] [b][i]"Some Thoughts on Problem Posing Sandboxes", mathMINDhabits[/i] [url]https://sites.google.com/site/mathmindhabits/[/url].[/b]
