Problem-Solving Techniques In and Out of the Classroom--By Kathy Jaqua

For many people, mathematics is about solving problems, but to mathematicians, mathematics is about problem-solving. Finding a path through problem solving can reveal how mathematics shows up in our lives every day. Follow these students along their paths through problem solving techniques as revealed in their mathematical selfies.
While there are lots of problem solving techniques that are specific to a particular type of problem, there are also some basic techniques that can be used for almost any problem. Students start to see these techniques early in school, but when faced with a problem to solve, they don’t always recall and use them. Yet, basic problem solving techniques are often used in day-to-day life. Let’s look at some mathematical selfies that illustrate three common problem-solving techniques: Draw a diagram, Make a systematic list, and Create subproblems.

Draw a diagram

 Because part of the goal of the problem-solving technique draw a diagram is to visualize information in a more holistic way, it is not surprising that drawing a map was a common illustration of a day-to-day use of this technique. In each of these mathematical selfies, the author is using the diagram to direct attention to important components of the map. In photo 1, the notion of scale does not seem to be important, only the relative placement of landmarks. Because the route is short, the lack of scale is possible without risk of misdirection.








In photo 2, however, part of the point of the map is to indicate the location of a particular point within the county. In this instance, scale is critical to make the map useful in illustrating the point of the story. Similarly in mathematics, sometimes diagrams need scale to aid in visualizing a problem as in investigations of similarity, while in other cases, such as the order of people standing in a line, scale is not critical. Recognizing that a diagram can be helpful in each case is a powerful problem-solving tool.

Make a systematic list

Lists are part of almost everyone’s life. It’s how we remember what groceries to pick up or the day’s agenda. These two types of lists, however, are different, and that difference is critical to good problem solving. A systematic list separates a way to randomly record a bunch of information and a way to use a list as an effective organizing tool. In each of these mathematical selfies, the authors show how a system for the list is the key to understanding the relationships within the information. In photo 3, the number of chairs needed for each table at a wedding reception is recorded based on the system of numbering the tables. I suspect there is also a diagram somewhere showing the relative locations of each of the numbered tables. This systematic list indicates how to separate identical items, the chairs, into groups based on table location. Photo 4 does not rely on an ordering that is connected to physical placement. Here the system used is to separate information by categories. The number of items within each category is not identical, and in many ways the items are not even comparable. These two ways to use systematic lists are reflected in mathematics. Early introduction to division is often based on separating identical objects into groups. The system used for that separation may be the number of groups or the size of each group. The two systems will yield different answers, and the relationship between those answers is one important connection between multiplication and division. Separating a list of items by categories shows up when we consider combinations. The categories do not have to have a particular order, as in types of clothes, and the items within each category do not need to be the same, as in color or type or size. But we can use a systematic list based on category and attribute to determine all the possible combinations.

Create subproblems

Breaking up tasks into smaller parts and completing each part makes large projects possible within our busy lives. If you want to redecorate a room, for example, you probably won’t try to do everything all at once. You will likely separate the job into smaller tasks that can be completed independently. The authors of these mathematical selfies show exactly that process of creating subproblems. In photo 5, the task of determining each person’s share of a bill is broken down into the various arithmetic tasks that will lead to the final solution. Here the subproblems must be completed in order as the information needed for each task comes from the solution to the previous one.

Photo 6 also shows the separation of a large task, completing all assignments for a week, into smaller subproblems; however, in this case, the tasks are independent. Assuming that the student is conscientious so that the due dates are not all in the week depicted, the individual assignments may be completed in a variety of orders without affecting the completion of the task as a whole. Similarly, in mathematics, completing a complex calculation requires the use of order of operations, which dictates a series of subproblems where the completion of each subproblem provides the basis for the next one and thus only an established order will yield the correct solution. In other problems, such as converting lbs/in2 into grams/cm2 we can use a series of unit conversions that may be applied in different orders and still produce the same overall solution. In each case, the use of subproblems makes the overall result achievable.

These mathematical selfies clearly depict problem-solving techniques in daily life. The comparison of these photos of problem-solving techniques to mathematical problems that students will see in a classroom illustrates the universal nature and value of problem-solving techniques both in and out of the classroom.

Three notions of angle -- By Axelle Faughn

If you stop for a moment to think about the notion of angle, you will realize that what seems like a pretty simple geometric figure of two lines (or rays) intersecting one another, actually has far-reaching ramifications in the world of mathematics. With the help of Mathematical Selfies, we consider three major interpretations of the concept of angles, which we believe must be presented to all children learning about angles in school.

1) Angle as sector

First let us explore the idea of angle as "space in-between two lines", in other words a sector of the plane. This is the original definition of angle which is introduced early on in schools when studying shapes. Children pretty quickly know how to differentiate right triangles from equilateral ones for instance by looking at the positions of the sides with respect to one another. When prompted to illustrate the notion of angles, this is often the one students consider since structures heavily rely upon how angles fit together within a plane.

Picture 1: Angle of best fit

Picture 2: Reaching out into the light

Picture 3: One fifth of 360

Picture 4: Angles in constructions

As in picture 1-4, we can assign a measurement to these angles and categorize them as acute, obtuse or right. Repeated practice with finding all three types of angles in the world around us seems like an important part of internalizing these notions for students in primary grades.

2) Angle as arc length

When more accurate angle measurements come into play, we step into the second notion of angle, which can be thought of as an arc length, or a curved measurement. The use of a protractor as a measuring device illustrates this notion quite nicely since the tool itself has the shape of the arc along which one will read the units.

This aspect of angles can be associated with circular motion and may come with a positive or a negative direction as in trigonometry. There is a very clear connections between arc length and angle measurement established through the definition of radians for measuring angles: if an angle Theta at a given vertex is measured in radians, then the arc length S spanned by this angle along a circle of radius R centered at that vertex is given by S=(Theta)x(R) . In trigonometry that is why in a circle of radius one unit, we can alternatively think of angles as sectors between two rays of the circle, or as distanced arc length of a point traveling on the circle (Picture 5).

Picture 5: The trigonometry of a fan

Picture 6: Copying an angle

Angles as circular motion are also used when copying an angle with the use of the traditional construction tools compass and straightedge (Picture 6). In systems of levers and pulleys, or just hinges on a door (Picture 7) where acute versus obtuse angles determine the closing versus opening of the door, these notions become strategic in gaining mechanical advantage, one of humans' everlasting endeavors.

Picture 7: Using angles for mechanical motion

3) Angle as walking motion 

When giving directions to someone you may find yourself telling them to make a 90 degree left or right turn, or explaining that the road takes a "sharp" right. All these, along with the idea of U-Turns, refer to angles that allow us to locate ourselves in the plane. Following a GPS uses the same types of commands. This notion of angles is less often considered in the mathematics classroom, although certainly not less important in our world of satellites and geo-location devices. Even Pokemon-Go wouldn't work without it! It is therefore worth mentioning here as using the idea of vector angles, in other words the angle established by two directions, wherever these originate from in the plane. Here we dive into the world of equivalence classes. In my geometry class, students have yet to submit illustrations for this particular angle notion, which I take as a hint that I do not emphasize it enough in class.

Picture 8: Octagon dilation

Interestingly enough, angles also play an important role in picture taking. Depending on the angle the photographer positions themselves with respect to their representations, these might change from the object to the selfie based on the angle from which the picture was taken. For instance in Picture 8 illustrating similarity, the viewer knows that a regular octagon is equiangular, however it is not so visually obvious if not facing the shape directly from below.

Finally, any adept photographer knows to use angles for capitalizing on the power of reflections to enhance a picture or create interesting patterns. Picture 9 is only one example of this play with reality using the laws of incidence.

Picture 9: Tree reflection

Picture 10 provides a much more creative illustration of using the mathematics of angles for the purpose of artistic ventures.

            

              Picture 10: Optical incidence, courtesy of Eric Mortemousque